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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I Yaw ^_\j ' p \ UNITS AND PHYSICAL CONSTANTS. __ \ '- UNITS AND PHYSICAL CONSTANTS. \ - \ ■•-. . \ \ £7^0 ' • r. UNITS AND PHYSICAL CONSTANTS. BY J. D. JgVERETT, M.A., D.O.L., F.R.S., F.R.S.E., PROFESSOR OF NATURAL PHILOSOPHY IN QUEEN'S COLLEGE, BELFAST. SECOND EDITION. \ |Cotti0n : MAOMILLAN AND CO., AND NEW YORK. , 1886. ^ [The riffht of transMion and r^produdioa -U rewrot-^A I I I! I THE NEW YORK I PUBLIC LIBRARY I 825920 1^ ASTOR. LENOX AND i'LDEN FOUNDATION.- 1918 L -iiTTrri OLASGOW : PRINTED AT THB UNIVERSITY PBE88 BT ROBERT MACLEHOSE. PREFACE TO "ILLUSTRATIONS OF THE O.G.S. SYSTEM OF UNITS," PUBLISHED BY THE PHYSICAL SOCIETY OF LONDON IN 1875. The quantitative study of physics, and especially of the relations between different branches of physics, is every day receiving more attention. To facilitate this study, by exemplifying the use of a system of units fitted for placing such relations in the clearest light, is the main object of the present treatise. A complete account is given of the theory of units ah initio. The Oentimetre-Gramme-Second (or O.G.S.) sys- tem is then explained ; and the remainder of the work is occupied with illustrations of its application to various branches of physics. As a means to this end, the most important experimental data relating to each subject are concisely presented on one uniform scale — a luxury hitherto unknown to the scientific calculator. I am indebted to several friends for assistance in special departments — ^but especially to Professor Clerk Maxwell and Professor G. 0. Foster, who revised the entire manu- script of the work in its original form. Great pains have been taken to make the work correct as a book of reference. Readers who may discover any errors will greatly oblige me by pointing them out. PREFACE TO FIJEIST EDITION OF UNITS AND PHYSICAL CONSTANTS, 1879. This Book is substantially a new edition of my *• Illus- trations of the C.G.S. System of Units " published in 1875 by the Physical Society of London, supplemented by an extensive collection of physical data. The title has been changed with the view of rendering it more generally intelligible. Additional explanations have been given upon some points of theory, especially in connection with Stress and Strain, and with Coefl&cients of Diffusion. Under the former head, I have ventured to introduce the terms "resilience" and "coefficient of resilience," in order to avoid the multiplicity of meanings which have become attached to the word "elasticity." A still greater innovation has been introduced in an extended use of the symbols and processes of multiplication and division, in connection with equations which express not numerical but physical equality. The advantages of this mode of procedure are illustrated by its application to the solution of the most difficult problems on units that I have been able to collect from standard text-books (chiefly from Wormell's * Dynamics '). I am indebted to several friends for contributions of experimental data. viii PREFACE TO FIRST EDITION. A Dutch translation of the first edition of this work has been made by Dr. C. J. Matthes, Secretary of the Boyal Academy of Sciences of Amsterdam, and was pub- lished in that city in 1877. Though the publication is no longer officially connected with the Physical Society, the present enlarged edition is issued with the Society's full consent and approval. PREFACE TO THE PRESENT EDITION. In collecting materials for this edition, I have gone care- fully through the Transactions and Proceedings of the Royal Society, the Royal Society of Edinburgh, and the Physical Society of London, from 1879 onwards, besides, consulting numerous papers, both English and foreign, which have been sent to me by their authors. I have also had the advantage of the co-operation of Dr. Pierre Chappuis (of the Bureau International des Poids et Mesures), who has for some years been engaged in pre- paring a German edition. Several items have been ex- tracted from the very elaborate and valuable Physikalisch- Gheinische Tabellen of Landolt and Bbrnstein (Julius Springer, Berlin, 1883). Among friends to whom I am indebted for data or useful suggestions, are Prof. Barrett, Mr. J. T. Bottomley, Prof. G. C. Foster, Prof. Lodge, Prof. Newcomb, Mr. Preece, and the Astronomer-Royal. The expository portions of the book are for the most part unchanged; but a Supplemental Section has been added (p. 34) on physical deductions from the dimensions of units ; a simplification has been introduced in the dis- cussion of adiabatic compression (p. 1 25) ; and the account of thermoelectricity (p. 172) has been re-written and enlarged. The name "thermoelectric height" has been introduced to denote the element usually represented by the ordinates of a thermoelectric diagram. X PREFACE TO THE PRESENT EDITION. The preliminary " Tables for reducing other measures to C.G.S. measures" have been greatly extended, and in each case the reciprocal factors are given which serve for reducing from C.G.S. measures to other measures. Pro- fessor Miller's comparison of the kilogramme and pound is supplemented by three later comparisons officially made at the Bureau International, A nearly complete list of the changes and additions now introduced is appended to this Preface, as it will probably be useful to possessors of the previous edition. The adoption of the Centimetre, Gramme, and Second, as the fundamental units, by the International Congress of Electricians at Paris in 1881, led to the immediate execution of a French translation of this work, which was published at Paris by Gauthier-Yillars in 1883. The German translation was commenced about the same time, but the desire to perfect its collection of physical data has caused much delay. It will be brought out by Ambrosius Barth, the publisher of Wiedemann's Annalen. A Polish edition, by Prof. J. J. Boguski, was published at Warsaw in 1885 ; and permission has been asked and granted for the publication of an Italian edition. J. D. EVERETT. Belfast, September, 1886. LIST OF CHANGES AND ADDITIONS. PAOEB Tables for conversion to and from C.G.S., - - - 1-4 Formula for gr, 26 " Watt " defined, 30 Physical deductions from dimensions, .... 34-37 Specific gravity table, 40 Surface tension of liquids, 49, 50 Thickness of soap films, 50, 51 Poisson's ratio, 62 Velocity of Hght, 76 Indices of refraction of crystals, etc. , - - - - 80, 81 Refraction and dispersion of gases, 83-85 Rotation by quartz, 85 Candle, carcel, etc., 86 Specific heat, 90-94 Melting, 95-97 Boiling, 98 Pressure of steam from 0° to 150*, 102 Critical points of gases, 103 Conductivity (thermal) of solids, - • - - 115, 116 „ ,, of liquids, 117 Joule's equivalent, 121 Adiabatic compression, 125, 126 Expansion of mercury, 129 Collected data for air, 129 Density of moist air, 130 Magnetic susceptibility, 133 Greenwich magnetic elements, 138 Magneto-optic rotation, 139 Ratio of the two units of electricity, .... 145 Specific inductive capacity, 148-I5ft xii CHANGES AND ADDITIONS. PA Practical units, 151, Resolutions of Congress and Conference, - • - 153, Resistance, 15^ Gauge and resistance of wires, 165. Electro-motive force of cells, 167, Thermoelectricity, 175 Electrochemic€il equivalents, - • - - - - 17S Heat of combination of cells, - ! - - - - 181, Compression of liquids, Expression of decimal multiples, etc., .... CONTENTS. PAGES Tables for Reducing to and from C.G.S. Measures, 1-4 Chapter L — General Theory of Units, - - - 5-18 Chapter II. — Choice of Three Fundamental Units, - 19-24 Chapter HI. — ^Mechanical Units, 25-34 Supplemental Section, on Physical Deductions from Dimensions, 34-37 Chapter IV.— Hydrostatics, 38-51 Chapter V. — Stress, Strain, and Resilience, • - 52-64 Chapter VI. — Astronomy, 65-69 Chapter VII.— Velocity of Sound, .... 70-74 Chapter Vin.— Light, 75-86 Chapter IX.— Heat, 87-130 Chapter X.— Magnetism, 131-139 Chapter XL —Electricity, 140-188 Omission and Suggestion, 189, 190 Appendix.— Reports of Units Cemmittee of British Association, - - 191-195 Index, 196-200 UNITS AND PHYSICAL CONSTANTS. TABLES FOR REDUCING TO AND FROM C.a.S. MEASURES. The abbreviation cm, is used for centimetre or centimetres, gm,, „ gramme or grammes^ c.c, „ cubic centimetre(8). The numbers headed " reciprocals " are the factors for reducing ^om C.G.S. measures. Length, cm. Reciprocals. 1 inch, - 2-5400 •39370 1 foot, - 30-4797 •032809 1 yard,- 91-4392 •010936 1 mile, = 160933 6-2138 xlO-« 1 nautical mile, - = 185230 5-398 xlO-« More exactly, according to Captain Clarke's compari- sons of standards of length (printed in 1866), the metre is equal to 1-09362311 yard, or 3-2808693 feet, or 39-370432 inches, the standard metre being taken as correct at 0° C, and the standard yard as correct at 16f'' C. Hence the inch is 2^5399772 centimetres. Area, 1 sq. inch, 1 sq. foot, 1 sq. yard, 1 sq. mile, i) sq. cm. Reciprocals. 6-4516 •1550 929 01 •001076 8361-13 •0001196 2-59xl0i« 3-861 X 10" A 2 UNITS AND PHYSICAL CONSTANTS. Volume. cub. cm. Reciprocals. I cubic inch, - -= 16-387 •06102 1 cubic foot, - - 28316- 3-532x10-6 1 cubic yard, - -764535- 1-308 xl0-« 1 pint, - - = 567-63 -001762 ] gallon, - = 4541- Mass, •0002202 gm. Reciprocals. 1 grain, - - -0647990 15-432 I ounce avoir.. - - 28-3495 •035274 1 pound ,, - =453-59 -0022046 1 ton, - ■ = 1-01605 xlO« 9-84206x10-7 According to the compaidson made by Professor W. H. Miller in 1844 of the "kilogramme des Archives," the standard of French weights, with two English pounds of platinum, and additional weights, also of platinum, the kilo- gramme is 15432-34874 grains, of which the new standard })Ound contains 7000. Hence the kilogramme would be 2-2046212 pounds, and the pound 453^59265 grammes. Three standard pounds, one of platinum-iridium and the other two of gilded bronze, belonging to the Standards Department^ were compared, in 1883, at the Bureau In- ternational des Poids et MesureSy with standards belong- ing to the Bureau, and their values in grammes were found to be respectively 453-59135, 453-58924, 45358738. — Travaux et Memoires, tome IV. Velocity, 1 foot per second, 1 statute mile per hour, 1 nautical mile per hour, 1 kilometre per hoar^ cm. per sec. = 30-4797 =44 704 =51 453 =27-777 Reciprocals. •032809 •022369 •019435 •036 TABLES. 3 Acceleration, cm. per sec. per sec. Reciprocal. 1 ft. per sec. per sec, - =30-4797 032809 JDensiti/, gm. per c.c. Reciprocals. ^T»JI^^!^'ilHr**"™ "M =1000013 -999987 maximum density, - - ) I lb. per cubic foot, - - = 016019 62*426 1 grain per cubic inch, - - := 003954 252-88 Stress (in gravitation measure), gm. per sq. cm. Reciprocals. 1 lb. per sq. foot, - = '48826 2 0481 1 lb. per sq. inch, - =:: 70-31 -014223 1 inch of mercury at| ^ 3^.53^ ..^^^^^ 30 inches „ ,, =1036-0 00096525 760 mm. „ „ =1033-3 -00096777 Surface Tension (in gravitation measure), gm. per cm. Reciprocals. 1 ipi-ain per linear inch, - = -02551 39-20 lib. „ foot, - =14-88 -06720 Work (in gravitation measure), gm.-cm. Reciprocals. . 1 foot-pound, - =13825 7-2331 xlO-^ 1 f oot-grain, - = 1 -976 50632 1 foot-ton, - - = 3097x107 6-494x10-* I kilogrammetre, - = 10** 10" ** Rate of Working (in gravitation msasure), gm.-cm. per sec. Reciprocals. 1 horse-power, - = 7 -604 x 10» 1 3151 x 10 - ^ 1 force-de-cheval, - = 7*5x106 1-3333x10-7 Heat (in gravitation measure), gm.-cm. Reciprocals. Igm. deg., - - =42400 2-36xl0-» 1 lb. deg. Cent., - = 1 -923 x 10^ 5-2 x lO-s 1 „ Fahr., - = 1068x107 9*36x10-8 The following reductions of gravitation measures to absolute measures are on the assumption that ^ = 981 : — UNITS AND PHYSICAL CONSTANTS. Weight of 1 gm. , - - = 1 kilogm., - = 1 tonDe, - = 1 ton, - — 1 cwt. , - • = 1 lb. avoir., - = 1 oz. 1 grain, Force (in absolute measure). Dynes. »» 981 9*81 X 10« 9-81 X 10» 9-97 X 10« 4-98x107 4-45 X 105 2-78 X 10* 63-57 1 poundal, = 13825 Reciprocals. •001019 1019xl0-« l-019xl0-» 1-003 xlO-» 2-008 X 10 -» 2-247 X 10 -« 3-697 X 10-« -01573 7-2333x10-* (The ratio of the poundal to the dyne is independent of g, ) Stress (in absolute measure). Dynes per sq. cm. 1 lb. per sq. foot, • = -479 lib. „ inch, - = 6-9x10* 1 gm. „ cm., - = -981 Ikilo. „ decim., - = 9*81 xlO» 1 cnv of mercury at 0°C., =13338' 76 „ „ „ = 1-0136x108 linch „ „ = 3-388x10* 30 „ ,. „ = 1-0163 xlO« >» )) »» Beciprocals. •00209 1-45x10-5 •00102 102x10-* •0000736 9-866 X 10-?^ 2-95x10-* 9-84x10-7 ' Surface Tension (in absolute msasure). Dynes per cm. Reciprocals. 1 gm. per linear cm., - - = 981 -00102 1 grain ,, inch, - - = 25 '04 lib. „ foot, - - =14600 6-85x10-* Work and Energy (in absolute measure). Ergs. Reciprocals. Icm. cm.,- - = 981 -001019 1 kilogrammetre, = 9 -81 x 10^ 1 -019 x 10-8 1 foot-pound, - = 1-356x107 7-37x10-8 1 foot-poundal, - =421390 2-3731 x 10-« (The ratio of the ft. -poundal to the erg is independent of g,) I joule - - = 107 ergs. Mate of Working (in absolute measure). Ergs per sec. =7-46x109 = 7-36x109 = lO'^ 1 horse-power, 1 force-de-cheval, - 1 watt, - - - . Heat (in absolute measure). Ergs. Igm. deg., - - - =4-2x10'^ 1 lb. deg. Cent. , - - = 1 -905 x lO^o 1 ,. Fahr., - - =1 058x1010 »» Reciprocals. 1-34x10-19 1-36x10-19 10-7 Reciprocals. 2-38x10-8 5-25x10-11 9-45x10-11 CHAPTER I. GENERAL THEORY OF UNITS. Units and Derived Units, 1. The numerical value of a concrete quantity is its ratio to a selected magnitude of the same kind, called tTie unit. Thus, if L denote a definite length, and I the unit length, •=- is a ratio in the strict Euclidian sense, and is V called the numerical value of L. The numerical value of a concrete quantity varies directly as the concrete quantity itself, and inversely as the unit in terms of which it is expressed. 2. A unit of one kind of quantity is sometimes defined by reference to a unit of another kind of quantity. For example, the unit of area is commonly defined to be the area of the square described upon the unit of length ; and the unit of volume is commonly defined as the volume of the cube constructed on the unit of length. The units of area and volume thus defined are called derived units, and are more convenient for calculation than indepen- dent units would be. For example, when the above 6 UNITS AND PHYSICAL CONSTANTS. [chap. definition of the unit of area is employed, we can assert that [the numerical value of] the area of any rectangle is equal to the product of [the numerical values of] its length and breadth ; whereas, if any other unit of area were employed, we should have to introduce a third factor which would be constant for all rectangles. 3. Still more frequently, a unit of one kind of quantity is defined by reference to two or more units of other kinds. For example, the unit of velocity is commonly defined to be that velocity with which the unit length would be described in the unit time. When we specify a velocity as so many miles per hour, or so many^^e^ per second, we in effect employ as the unit of velocity a mile per hour in the former case, and a foot per second in the latter. These are derived units of velocity. Again, the unit acceleration is commonly defined to be that acceleration with which a unit of velocity would be gained in a unit of time. The unit of acceleration is thus derived directly from the units of velocity and time, and therefore indirectly from the units of length and time. 4. In these and all other cases, the practical advantage of employing derived units is, that we thus avoid the intro- duction of additional factors, which would involve needless labour in calculating and difficulty in remembeiing.* 5. The correlative term to derived is fundamental. Thus, when the units of area, volume, velocity, and * An example of such needless factors may be found in the rules commonly given in English books for findmg the mass of a body when its volume and material are given. ** Multiply the volume in cubic feet by the specific gravity and by 62*4, and the product will be the mass in pounds ; " or ** multiply the volume in cubic I.] GENERAL THEORY OF UNITS. 7 acceleration are defined as above, the units of length and time are called the fundamental units. Dimensions, 6. Let us now examine the laws according to which derived units vary when the fundamental units are changed. Let V denote a concrete velocity such that a concrete length L is described in a concrete time T ; and let v, Z, t denote respectively the unit velocity, the unit length, and the unit time. The numerical value of V is to be equal to the numerical value of L divided by the numerical value of T. But these numerical values are — , — , -: V I t hence we must have l^h' (1) This equation shows that, when the units are changed (a change which does not affect V, L, and T), v must vary directly as I and inversely &s t; that is to say, the unit of velocity varies directly as the unit of lengthy and inversely as the unit of time. Y Equation (1) also shows that the numerical value — of V a given velocity varies inversely as the unit of lengthy and directly as the unit of time, inches by the specific gravity and by 253, and the product will be the mass in grains." The factors 62*4 and 253 here employed would be avoided — that is, would be replaced by unity, if the unit volume of water were made the unit of mass. 8 UNITS AND PHYSICAL CONSTANTS. [chap. 7. Again, let A denote a concrete acceleration such that the velocity V is gained in the time T', and let a denote the unit of acceleration. Then, since the numerical value of the acceleration A is the numerical value of the velocity V divided by the numerical value of the time T, we have It t V But by equation (1) we may write —for—. We I T V thus obtain A^L t^ t a~JfT' (2) This equation shows that when the units a, l^ t are changed (a change which will not affect A, L, T or T'), a must vary directly as l^ and inversely in the duplicate ratio of t ; and the numerical value — will vary inversely a as I, and directly in the duplicate ratio of t In other words, the unit of acceleration varies directly as the unit of lengthy and inversely as tJie square of the unit of time; and the numerical value of a given acceleration varies inversely as the vm,it of lengthy and directly as the square of the unit of time. It will be observed that these have been deduced as direct consequences from the fact that [the numerical value of] an acceleration is equal to [the numerical value of] a length, divided by [the numerical value of] a time, and then again by [the numerical value of] a time. The relations here pointed out are usually expressed by 1.] GENERAL THEORY OF UNITS. 9 lArtrrfVi saving that the dimensions of acceleration* are .-r-^ — , or (time)- that the dimensions o/t/ie unit of acceleraiion* are unit of length (unit of time)2' 8. We have treated these two cases very fully, by way of laying a firm foundation for much that is to follow. We shall hereafter use an abridged form of reasoning, «uch as the following : — , .. length velocity = — r^- : •^ time ' , .. velocity length acceleration = — ; ^ = , . .^ » time (time)2 Such equations as these may be called dimensional eqvuitions. Their full interpretation is obvious from what precedes. In all such equations, constant numerical factors can be discarded, as not affecting dimensions. 9. As an example of the application of equation (2) we shall compare the unit acceleration based on the foot and second with the unit acceleration based on the yard and minute. Let I denote a foot, L a yard, t a second, T a minute, Ty a minute. Then a will denote the unit acceleration based on the foot and second, and A will denote the unit ♦Professor James Thomson (*Brit. Assoc. Report,* 1878, p. 452) objects to these expressions, and proposes to substitute the following : — "Change-ratio of unit of acceleration =;^^^j^^/' This is very clear and satisfactory as a full statement of the meaning intended ; but it is necessary to tolerate some abridg- ment of it for practical working. 10 UNITS AND PHYSICAL CONSTANTS. [chap, acceleration based on the yaid and minute. Equation (2) becomes a 1 "" V60/ 1200 ' ^ ^ that is to say, an acceleration in which a yard per minute of velocity is gained per minute, is A^ of an acceleration in which a foot per second is gained per second. Meaning of ^^per" 10. The word per, which we have several times em- ployed in the present chapter, denotes division of the quantity named before it by the quantity named after it. Thus, to compute velocity in feet per second, we must divide a number of feet by a number of seconds.* If velocity is continuously varying, let x be the number of feet described since a given epoch, and t the number doc of seconds elapsed, then - is what is meant by the at number of feet per second. The word should never be employed in the specification of quantities, except when the quantity named before it varies directly as the quantity named after it, at least for small variations — as, in the above instance, the distance described is ultimately pro- portional to the time of describing it. Extended Sense of the terms ** Multiplication " and " Division" 11. In ordinary multiplication the multiplier is always- * It is not correct to speak of interest at the rate of Fim Pounds per cent. It should be simply Five per cent. A rate of five pounds in every hundred pounds is not different from a rate of five- shillings in every hundred shillings. I.] GENERAL THEORY OF UNITS. H a mere numerical quantity, and the product is of the same nature as the multiplicand. Hence in ordinary division either the divisor is a mere numerical quantity and the quotient a quantity of the same nature as the dividend ; or else the divisor is of the same nature as the dividend^ and the quotient a mere numerical quantity. But in discussing problems relating to units, it is con- venient to extend the meanings of the terms " multiplica- tion " and " division." A distance divided by a time will denote a velocity — the velocity with which the given distance would be described in the given time. The dis- tance can be expressed as a unit distance multiplied by a numerical quantity, and varies jointly as these two factors ; the time can be expressed as a unit time multiplied by a numerical quantity, and is jointly proportional to these two factors. Also, the velocity remains unchanged when the time and distance are both changed in the same ratio. 1 2. The three quotients 1 mile ' 5280 ft. 22 ft. I hour 3600 sec' 15 sec. all denote the same velocity, and are therefore to be regarded as equal. In passing from the first to the second, we have changed the units in the inverse ratio to their numerical multipliers, and have thus left both the distance and the time unchanged. In passing from the second to the third, we have divided the two numeri- cal factors by a common measure, and have thus changed the distance and the time in the same ratio. A change in either factor of the numerator will be compensated by a proportional change in either factor of the denom- inator. 1 2 UNITS AND PHYSICAL CONSTANTS. [chap. 09 £«. ^ 22 Further, since the velocity - ~ — - is — of the velo- ID sec. 10 .. 1 ft. ^.^, , ^ ..22 ft. 22 ft. <;ity , we are entitled to write = 7^ . , 1 sec. 15 sec. 15 sea thus separating the numerical part of the expression from the units part. In like manner we may express the result of Art. 9 by writing yard _ 1 foot (minute)2 " 1200 * (second)^' Such equations as these may be called *' physical -equations," inasmuch as they express the equality of physical quantities, whereas ordinary equations express the equality of mere numerical values. The use of physical equations in problems relating to units is to be strongly recommended, as affording a natural and easy clue to the necessary calculations, and especially as obviating the doubt by which the student is often -embarrassed as to whether he ought to multiply or divide. 13. In the following examples, which illustrate the use of physical equations, we shall employ I to denote the unit length, m the unit mass, and t the unit time. Ex. 1. If the yard be the unit of length, and the acceleration of gravity (in which a velocity of 32*2 ft. per «ec. is gained per sec.) be represented by 2415, find the unit of time. We have I = yard, and 32-2 J^, = 2415 { = 2415 ^' (sec.)2 fi t^ .-. t^ = ^^1^ sec.* = 225 secS t = 15 sec. I.] GENERAL THEORY OP UNITS. 15 Ex. 2. If the unit time be the second, the unit density 162 lbs. per cub. ft., and the unit force* the weight of an ounce at a place where g (in foot-second units) is 32^ what is the unit length ? We have < = sec, J. = 162^3, and -^ = 32 . ?5l^, or ml^^ 32 oz. ft. = 2 lb. ft. sec* sec. Hence by division ^* = ^ (ft.)*. ? = J ft. = 4 in. Ex. 3. If the area of a field of 10 acres be represented by 100, and the acceleration of gravity (taken as 32 foot- second units) be 58 1, find the unit of time. We have 48400 (yd.)^=100 V, whence Z = 22 yd.; and 32-A. = 58| -4 = ^ ^> (sec.)^ ^ f 3 t' ' whence f = — — sec.^ =121 sec.^ ^=11 sec, Ex. 4. If 8 ft. per sec, be the unit velocity, and the acceleration of gravity (32 foot-second units) the unit acceleration, find the units of length and time. We have the two equations I Q it. L QO it. t sec. r sec* whence by division t = \ sec, and substituting this value of ^ in the first equation, we have 4 ?= 8 ft., Z= 2 ft. Ex. 6. If the unit force be 100 lbs, weight, the unit length 2 ft., and the unit time \ sec, find the unit mass, the acceleration of gravity being taken as 32 foot-second units. * For the dunensions of density and force, see Art 14. 14 UNITS AND PHYSICAL CONSTANTS. [chap. We have 1 = 2 ft., < = J see., ft. ml m 2 it. TOO lb. 32 sec." V TW" sec. tliat is 100 X 32 lb. =^2m,m = 100 lb. Ex. 6. The number of seconds in the unit of time is vqual to the number of feet in the unit of length, the unit of force is 750 lbs. weight [g being 32], and a cubic foot of the standard substance [substance of unit density] con- tains 13500 oz. Find the unit of time. Let t = x sec, then l = xft; also let m — y lb. Then we have ml _ y lb. xii. _ y lb. ft . __ --^ 09 lb. ft. t^ ^ sec.* X sec.*^ sec* or V- = 750 X 32. X whence \ = 13500 x }-, a^ 16 Hence by division 2 _ 750 X 32 X 16 __ 16* _ 16 , _ 16 ^ 13500 3^' "^ - T* ' " T'^"^ Ex. 7. When an inch is the unit of length and t the unit of time, the measure of a certain acceleration is ay when 5 ft. and 1 min. are the units of length and time respectively, the measure of the same acceleration is 10 a. Find t Equating the two expressions for the acceleration, we , inch TA 5 ft. have a --^ = 10 a . , t^ (mm.)* i.] GENERAL THEORY OF UNITS. 15 I j^ t ' \'> inch (min.)*'' ^ / v., whence t' = (min.)- ,^ . = J- = 6 (sec.)'' ^ ' 50 ft. 600 ^ ^ < = x/6 sec. Ex. 8. The numerical value of a certain force is 56 when the pound is the unit of mass, the foot the unit of length, and the second the unit of time ; what will be the numerical value of the same force when the hundredweight is the unit of mass, the yard the unit of length, and the minute the unit of time ? Denoting the required value by x we have p.« lb. ft. _ cwt. yard 00 — — X : — - — ; sec.- mm.'* a; = 56 ^^- ^*- (min.y sec./ cwt. yd. = 56 X -1- X 1 X 60^ = 600. ^ 112 3 Dimensions of Mechanical and Geometrical Quantities. 14. In the following list of dimensions, we employ the letters L, M, T as abbreviations for the words Length, MasSj Time, The symbol of equality is used to denote sameness of dimensions. Area = L^, Volume = L^, Velocity = -, Acceleration = =5, Momentum = -— . M Density = =-, density being defined as mass per unit 1-r volume. Force = -=^, since a force is measured by the momen- tum which it generates per unit of time, and is therefoi'e 16 UNITS AND PHYSICAL CONSTANTS. [chap, the quotient of momentum by time — or since a force i& measured by the product of a mass by the acceleration generated in this mass. Work = -7=5-, being the product of force and distance. ML* Kinetic energy - -7p^, being half the product of mass by square of velocity. The constant factor J can be omitted, as not affecting dimensions. ML* Moment of couple = -^pj , being the product of a force by a length. The dimensions of angle,* when measured by — - — , radius are zero. The same angle will be denoted by the same number, whatever be the unit of length employed. In fact we have — r^— = ~ = Ifi, radius L The work done by a couple in turning a body through any angle, is the product of the couple by the angle. The identity of dimensions between work and couple is thus verified. Angular velocity - -. Angular acceleration = — . Moment of inertia = ML*. ML* Angular momentum = moment of momentum = , * The name radian has been given to the angle whose arc is equal to radius. "An angle whose value in circular measure is 0** iB^an angle of radians.** 1.] GENERAL THEORY OF UNITS. 17 being the product of moment of inertia by angular velo- city, or the product of momentum by length. Intensity of pressurej or intensity of stress generally, being force per unit of area, is of dimensions ; that area . M IS Intensity of force of attraction at a point, often called simply force at a point, being force per unit of attracted mass, is of dimensions or =5. It is numerically mass T^ equal to the acceleration which it generates, and has accordingly the dimensions of acceleration. The absolute force of a centre of attraction, better called the strength of a centre, may be defined as the intensity of force at unit distance. If the law of attraction be that of inverse squares, the strength will be the product of the intensity of force at any distance by the square of this Curvature (of a curve) = — , being the angle turned by the tangent per unit distance travelled along the curve. Tortuosity = — , being the angle turned by the osculat- ing plane per unit distance travelled along the curve. The solid angle or aperture of a conical surface of any form is measured by the area cut off by the cone from a sphere whose centre is at the vertex of the cone, divided by the square of the radius of the sphere. Its dimensions are therefore zero ; or a solid angle is a numerical quan- tity independent of the fundamental units. B 18 UNITS AND PHYSICAL CONSTANTS, [chap. i. The specific cv/rvatvAre of a surface at a given point (Grauss's measure of curvature) is the solid angle de- scribed by a line drawn from a fixed point parallel to the normal at a point which travels on the surface round the given point, and close to it, divided by the very small area thus enclosed. Its dimensions are therefore — .. The mean curvature of a surface at a given point, in the theory of Capillarity, is the arithmetical mean of the curvatures of any two normal sections normal to each other. Its dimensions are therefore -=-. Li 19 CHAPTER II. CHOICE OF THREE FUNDAMENTAL UNITS. 15. Nearly all the quantities with which physical science deals can be expressed in terms of three funda- mental units ; and the quantities commonly selected to serve as the fundamental units are a definite length, a definite mass, a definite interval of time. This particular selection is a matter of convenience rather than of necessity ; for any three independent units are theoretically sufficient. For example, we might em- ploy as the fundamental units a definite mass, a definite amount of energy, a definite density. 16. The following are the most important considera- tions which ought to guide the selection of fundamental units : — (1) They should be quantities admitting of very accurate comparison with other quantities of the same kind. 20 UNITS AND PHYSICAL CONSTANTS. [chap. (2) Such comparison should be possible at all times. Hence the standards must be permanent — that is, not liable to alter their magnitude with lapse of time. (3) Such comparisons should be possible at all places. Hence the standards must not be of such a nature as to change their magnitude when carried from place to place. (4) The comparison should be easy and direct. Besides these experimental requirements, it is also desirable that the fundamental units be so chosen that the definition of the various derived units shall be easy, and their dimensions simple. 17. There is probably no kind of magnitude which so completely fulfils the four conditions above stated as a standard of masSy consisting of a piece of gold, platinum, or some other substance not liable to be affected by atmospheric influences. The comparison of such a standard with other bodies of a])proximately equal mass is effected by weighing, which is, of all the operations of the laboratory, the most exact. Very ac- curate copies of the standard can thus be secured; and these can be carried from place to place with little risk of injury. The third of the requirements above specified forbids the selection of a force as one of the fundamental units. Forces at the same place can be very accurately measured by comparison with weights; but as gravity varies from place to place, the force of gravity upon a piece of metal, or other standard weight, changes its magnitude in travelling from one place to another. A spring-balance, it is true, gives a direct measure of n.] THKEE FUNDAMENTAL UNITS. 21 force; but its indications are too rough for purposes of accuracy. 18. Length is an element which can be very accurately measured and copied. But every measuring instrument is liable to change its length with temperature. It is therefore necessary, in defining a length by reference to a concrete material standard, such as a bar of metal, to state the temperature at which the standard is correct. The temperature now usually selected for this purpose is that of a mixture of ice and water (0° C), observation having shown that the temperature of such a mixtiire is constant. The length of the standard should not exceed the length of a convenient measuring-instrument ; for, in comparing the standard with a copy, the shifting of the measuring- instrument used in the comparison introduces additional risk of error. In endrstanda/rdSy the standard length is that of the bar as a whole, and the ends are touched by the instrument every time that a comparison is made. This process is liable to wear away the ends and make the standard false. In Ime-standcurds, the standard length is the distance be- tween two scratches, and the comparison is made by optical means. The scratches are usually at the bottom of holes sunk halfway through the bar. 19. Time is also an element which can be measured with extreme precision. The direct instruments of mea- surement are clocks and chronometers; but these are checked by astronomical observations, and especially by transits of stars. The time between two successive tran- sits of a star is (very approximately) the time of the 22 UNITS AND PHYSICAL CONSTANTS. [chap. earth's rotation on its axis ; and it is upon the uniformity of this rotation that the preservation of our standards of time depends. Necessity for a Common Scale, 20. The existence of quantitative correlations between the various forms of energy, imposes upon men of science the duty of bringing all kinds of physical quantity to one common scale of comparison. Several such measures (called absolute measures) have been published in recent years; and a comparison of them brings very promi- nently into notice the great diversity at present existing in the selection of particular units of length, mass, and time. Sometimes the units employed have been the foot, the grain, and the second ; sometimes the millimetre, milli- gramme, and second ; sometimes the centimetre, gramme, and second; sometimes the centimetre, gramme, and minute ; sometimes the metre, tonne, and second ; some- times the metre, gramme, and second ; while sometimes a mixture of units has been employed ; the area of a plate, for example, being expressed in square metres, and its thickness in millimetres. A diversity of scales may be tolerable, though undesir- able, in the specification of such simple matters as length, area, volume, and mass when occurring singly ; for the reduction of these from one scale to another is generally understood. But when the quantities specified involve a reference to more than one of the fundamental units^ and especially when their dimensions in terms of these units are not obvious, but require 'careful working out. n.] THREE FUNDAMENTAL UNITS, 23 there is great increase of difficultj and of liability to mistake. A general agreement as to the particular units of length, mass, and time which shall be employed — if not in all scientific work, at least in all work involving complicated references to units — is urgently needed ; and almost any one of the selections above instanced would be better than the present option. 21. We shall adopt the recommendation of the Units Committee of the British Association (see Appendix), that all specifications be referred to the Centimetre^ the Gramme, and the Second. The system of units derived from these as the fundamental units is called the C.G,S. system; and the units of the system are called the CG.S, units. The reason for selecting the centimetre and gramme, rather than the metre and gramme, is that, since a gramme of water has a volume of approximately 1 cubic centimetre, the former selection makes the density of water unity; whereas the latter selection would make it a million, and the density of a substance would be a million times its specific gravity, instead of being identical with its specific gravity as in the CG.S. system. Even those who may have a preference for some other units will nevertheless admit the advantage of having a variety of results, from various branches of physics, re- duced from their original multiplicity and presented in one common scale. 22. The adoption of one common scale for all quan- tities involves the frequent use of very large and very 24 UNITS AND PHYSICAL CONSTANTS, [chap. ii. small numbers. Such numbers are most conveniently written by expressing them as the product of two factors, one of which is a power of 10 ; and it is usually advan- tageous to effect the resolution in siich a way that the exponent of the power of 10 shall be the characteristic of the logarithm of the number. Thus 3240000000 will be written 3-24 x 10», and -00000324 will be written 3-24 X 10-«. 26 CHAPTER III. MECHANICAL UNITS. Value of g. 23. AccELEBATiON is defined as the rate of increase of velocity per unit of time. The C.G.S. unit of accelera- tion is the acceleration of a body whose velocity increases in every second by the C.G.S. unit of velocity — namely, by a centimetre per second. The apparent acceleration of a body falling freely under the action of gravity in vacuo is denoted by g. The value of ^r in C.G.S. units at any part of the earth's surface is approximately given by the following formula, g = 980-6056 - 2-5028 cos 2 A - -OOOOOSA, A. denoting the latitude^ and h the height of the station (in centimetres) above sea-level. The constants in this formula have been deduced from numerous pendulum experiments in different localities, the length I of the seconds' pendulum being connected with the value of g by the formula g = ttH, Dividing the above equation by tt^ we have, for the length of the seconds' pendulum, in centimetres, /= 99-3562 - -2536 cos 2 A - -OOOOOOSA. 26 UNITS AND PHYSICAL CONSTANTS. [chap. At sea-level these formulse give the following values for the places specified : — Latitude. Value of g. Value of U « Equator, - 6 978 10 99-103 Latitude 45°, - 45 980*61 99*356 Munich, - 48 9 980-88 99-384 Paris, - 48 50 980-94 99-390 Greenwich, 61 29 981-17 99-413 Gottingen, 51 32 981-17 99-414 Berlin, 52 30 981-25 99-422 Dublin, - 53 21 981-32 99-429 Manchester, 53 29 981-34 99-430 Belfast, - 54 36 981-43 99-440 Edinburgh, 55 57 981-54 99-451 Aberdeen, 57 9 981-64 99-461 Pole, 90 983-11 99-610 The difference between the greatest and least values (in the case of both g and I) is about —— of the mean 196 value. 24. The Standards Department of the Board of Trade, being concerned only with relative determinations, has adopted the formula (7 = 5r,(l--00257cos2A)A-|A\ k denoting the latitude, h the height above sea-level, K the earth's radius, g^ the value of g in latitude 45 ** at sea- level, which may be treated as an unknown constant multiplier. Putting for It its value in centimetres, the formula gives g^g^{\ - -00257 cos 2A - 1-96A x lO"*), where h denotes the height in centimetres. in.] MECHANICAL UNITS. 27 The formula which we employed in the preceding section gives ^=^,(1 - -00255 cos 2A)(l - ^V As regards the factor dependent on height, theory indi- cates 1 - ~ as its correct value for such a case as that of Xv a balloon in mid-air over a low-lying country ; the value 1 - J :g- may be accepted as more correct for an elevated plateau on the earth's surface. Force, 25. The C.G.S. unit of force is called the dyne. It is the force which, acting upon a gramme for a second, generates a velocity of a centimetre per second. It may otherwise be defined as the force which, acting upon a gramme, produces the C.G.S. unit of acceleration, or as the force which, acting upon any mass for 1 second, produces the C.G.S. unit of momentum. To show the equivalence of these three definitions, let m denote mass in grammes, v velocity in centimetres per second, t time in seconds, F force in dynes. Then, by the second law of motion, we have 1 .. force acceleration = ; mass P that is, if a denote acceleration in C.G.S. units, a= - ; m hence, when a and m are each unity, F will be unity. Again, by the nature of uniform acceleration, we have v-at, V denoting the velocity due to the acceleration a, continuing for time t. 28 UNITS AND PHYSICAL CONSTANTS. [cfHAP. Hence we have F = ma = — . Therefore, if mv = 1 t and t=ly we have F= 1. As a particular case, if in -I, v = l, t = l, we have F = l. 26. The force represented by the weight of a gramme varies from place to place. It is the force required to sustain a gramme in vacuo, and would be nil at the earth's centre, where gravity is nil. To compute its amount in dynes at any place where g is known, observe that a mass of 1 gramme falls in vacuo with acceleration g. The force producing this acceleration (namely, the weight of the gramme) must be equal to the product of the mass and acceleration, that is, to g. The weight (when weight means force) of 1 gramme is therefore g dynes ; and the weight of m grammes is mg dynes. 27. Force is said to be expressed in gravitation-measure when it is expressed as equal to the weight of a given mass. Such specification is inexact unless the value of g is also given. For purposes of accuracy it must always be remembered that the pound, the gramme, etc., are, strictly speaking, units of mass. Such an expression as " a force of 100 tons " must be understood as an abbrevia- tion for " a force equal to the weight [at the locality in question] of 100 tons." 28. The name poundal has recently been given to the unit force based on the pound, foot, and second ; that is, the force which, acting on a pound for a second, gene- rates a velocity of a foot per second. It is — of the HI.] MECHANICAL UNITS. 29 weight of a pound, g denoting the acceleration due to gravity expressed in foot-second units, which is about 32*2 in Great Britain. To compare the poundal with the dyne, let x denote the number of dynes in a poundal ; then we have gm. cm. __ lb. ft. X sec. 2 sec. 2 x^— , — = 453-59 X 30-4797 = 13825. gm. cm. Work mui Energy, 29. The C.G.S. unit of work is called the erg. It is the amount of work done by a dyne working through a distance of a centimetre. The C.G.S. unit of energy is also the erg, energy being measured by the amount of work which it represents. 30. To establish a rule for computing the kinetic energy (or energy due to the motion) of a given mass moving with a given velocity, it is sufficient to consider the case of a body falling in vacuo. When a body of m grammes falls through a height of /* centimetres, the working force is the weight of the body — ^that is, gTTh dynes, which, multiplied by the distance worked through, gives gmh ergs as the work done. But the velocity acquired is such that «?- = 2^A. Hence we have gmh = ^v^. The kinetic energy of a mass of m grammes moving with a velocity of v centimetres per second is therefore i^v^ ergs ; that is to say, this is the amount of work which would be required to generate the motion of the body, or is the amount of work which the body 30 UNITS AND PHYSICAL CONSTANTS. [chap. would do against opposing forces before it would come to rest. 31. Work, like force, is often expressed in gravitation- measv/re. Gravitation units of work, such as the foot- pound and kilogramme-metre, vary with locality, being proportional to the value of g. One gramme-centimetre is equal to g ergs. One kilogramme-metre is equal to 100,000 g ergs. One foot-poundal is 463-69 x (30-4797)^ = 421390 ergs. One foot-pound is 13,825 g ergs, which, if g be taken as 981, is 1-356x10" ergs. 32. The C.G.S. unit rate of working is 1 erg per second. Watt's " horse-power " is defined as 560 foot-pounds per second. This is 7*46 x 10^ ergs per second. The ** force de cheval " is defined as 75 kilogrammetres per second. This is 7*36 X 10^ ergs per second. We here assume ^ = 981. A new unit of rate of working has been lately intro- duced for convenience in certain electrical calculations. It is called the Watt^ and is defined as 10*^ ergs per second. A thousand watts make a kilowatt The following tabular statement will be useful for reference. 1 Watt = 10*^ ergs per second = -00134 horse-power = '737 foot-pounds per second = -101 9 kilogram- metres per second. 1 Kilowatt = 1 -34 horse-power. 1 Horse-power = 550 foot-pounds per second = 76*0 kilogrammetres per second = 746 watts = 1 -01 385 force de cheval. 1 Force de cheval = 75 kilogrammetres per second = 642 -48 foot-pounds per second =736 watts = -9863 horse-power. ni.l MECHANICAL UNITS. 31 Examples, 1. If a spring balance is graduated so as to show the masses of bodies in pounds or grammes when used at the equator, what will be its error when used at the poles, neglecting effects of temperature ? Ana, Its indications will be too high by about y^^ of the total weight. 2. A cannon-ball, of 10,000 grammes, is discharged with a velocity of 45,000 centims. per second. Find its kinetic energy. Ana, I X 10000 x (46000)* = 1*0125 x lO^^ ergs. 3. In last question find the mean force exerted upon the ball by the powder, the length of the barrel being 200 centims. Ana. 5-0625 x lO^^ dynes. 4. Given that 42 million ergs are equivalent to 1 gramme-degree of heat, and that a gramme of lead at 10° 0. requires 15*6 gramme-degrees of heat to melt it; find the velocity with which a leaden bullet must strike a target that it may just be melted by the collision, suppos- ing all the mechanical energy of the motion to be converted into heat and to be taken up by the bullet. We have Jt;*=15'6xJ, where J = 42x10^. Hence »*=1310 millions; i?-36'2 thousand centims. per second. 5. With what velocity must a stone be thrown verti- cally upwards at a place where ^ is 981 that it may rise tea height of 3000 centims.? and to what height would it ascend if projected vertically with this velocity at the surface of the moon, where ^ is 150 ? Ana, 2426 centims. per second ; 19620 centims. 32 UNITS AND PHYSICAL CONSTANTS. [chap. Centrifugal Force, 33. A body moving in a curve must be regarded as continually falling away from a tangent The accelera- v tion with which it falls away is — , i? denoting its velocity and r the radius of curvature. The acceleration of a body in any direction is always due to force urging it in that direction, this force being equal to the product of mass and acceleration. Hence the normal force on a body of m grammes moving in a curve of radius r centimetres, with velocity v centimetres per second, is dynes. This force is directed towards the centre of curvature. The equal and opposite force with which the body reacts is called centrifugal force. If the body moves uniformly in a circle, the time 27n- of revolution being T seconds, we have v = ~ifr'i v^ /27r\2 hence "^("rfrj^j aiid the force acting on the body is /27r\2 //*n rn-j dynes. If n revolutions are made per minute, the value of T is — , and the force is mr ('Sj ^^^ Examples. 1. A body of m grammes moves uniformly in a circle of radius 80 centims., the time of revolution being ^ of a in.] MECHANICAL UNITS. 33 second. Find the centi'ifugal force, and compare it with the weight of the body. -^1 X 80 = m X 647r* X 80 = 50532 m dynes. The weight of the body (at a place where ^ is 981) is 981 m dynes. Hence the centrifugal force is about b2\ times the weight of the body. 2. At a bend in a river, the velocity in a certain part of the surface is 170 centims. per second, and the radius of curvature of the lines of flow is 9100 centims. Find the slope of the surface in a section transverse to the lines of flow. Ana, Here the centrifugal force for a gramme of the water is 0^=3-176 dynes. If ^ be 981 the slope will be = -r^ ; that is, the surface will slope upwards 981 o09 from the concave side at a gradient of 1 in 309. The general rule applicable to questions of this kind is that the resultant of centrifugal force and gravity must be normal to the surface. 3. An open vessel of liquid is made to rotate rapidly round a vertical axis. Find the number of revolutions that must be made per minute in order to obtain a slope of 30° at a part of the surface distant 10 centims. from the axis, the value of ^ being 981. f Ans, We must have tan 30° = ^, where /* denotes the 9 intensity of centrifugal force — that is, the centrifugal force per unit mass. We have therefore c 34 UNITS AKD PHYSICAL CONSTANTS. [chap. 981 tan 30° = lo(|^Y' ** **«'^°*»°8 w denoting the number of revolutions per minute, o o 90 Hence n = 71'9. 4. For the intensity of centrifugal force at the equator due to the earth's rotation, we have r = earth's radius = 6*38 X 10^, T = 86164, being the number of seconds in a sidereal day. •■• ^'"-i^i)-^-^^- This is about ^— of the value of g. If the earth were at rest, the value of g at the equator would be greater than at present by this amount. If the earth were revolving about 1 7 times as fast as at present, the value of g at the equator would be nil. Supplemental Section. On the help to he derived from Dimensions in investi- gating Physical Formulce, When one physical quantity is known to vary as some power of another physical quantity, it is often possible to find the exponent of this power by reasoning based on dimensions, and thus to anticipate the results — or some of the results — of a dynamical investigation. Examples, 1. The time of vibration of a simple pendulum in a small arc depends on the length of the pendulum and the intensity of gravity. If we assume it to vary as the wi* HL] MECHANICAL UNITS. 35 power of the length, and as the ri'* power of g, and to be independent of everything else, the dimensions of a time must equal the m** power of a length, multiplied by the w'* power of an acceleration, that is T = L"*(LT-2)" = L*" L** T-2" __ T »» + n T^-2n Since the dimensions of both members are to be identical, we have, by equating the exponents of T, 1 = - 2^1, whence n= - ^, and by equating the exponents of L, m + n = 0, whence m = J ; that is, the time of vibration varies directly as the square root of the length, and inversely as the square root of ^. 2. The velocity of sound in a gas depends only on the density D of the gas and its coefficient of elasticity E, and we shall assume it to vary as D"* E". The dimensions of velocity are LT"\ The dimensions of density, or — = , are ML~'. volume The dimensions of E, which will be explained in the chapter on stress and strain, are , or (MLT"^)L~^, or area ML-i T-l The equation of dimensions is _. IJTm+n T —3m—n rp— 2n whence, by equating coefficients, we have the three equations 1 = - 3m - 71, - 1 = - 2n, m + n = 0, to determine the two unknowns m and n. 36 UNITS AND PHYSICAL CONSTANTS. [c The second equation gives at once w = J. The third then gives m= -J, and these values will be found to satisfy the first equs also. The velocity, then, varies directly as the square ro< E, and inversely as the square root of D. 3. The frequency of vibration f for a musical st (that is, the number of vibrations per unit time) dep on its length I, its mass niy and the force with whic is stretched F. The dimensions of/ are T"\ „ „ X „ XU.XJJ. . Assume that/ varies as I'm^ F'. Then we have T-^ = L*M»'M^L*T-^ = L'+'M»'+*T-% giving -l=-20, x + z = 0, y + z=^0; whence ^ = h ^= "h 1/-"^' Hence /varies as ^ /.;— . 4. The angular acceleration of a uniform disc roun( axis depends on the applied couple G, the mass of the M, and its radius E. Assume it to vary as G* M*' R*. The dimensions of angular acceleration are T"-^. G „ ML^T- Hence we have T-2 = M* W T-^ M*' L\ irr.] MECHANICAL UNITS. 37 gi^v^ing "2= -2xj re + 1/ = 0, 2x + z = 0, w-lience 05 = 1, y=— 1, «=-2. Hence the angular acceleration varies as . In the following example the information obtained is less complete : — 5. The range of a projectile on a horizontal plane through the point of projection depends on the initial velocity V, the intensity of gravity g, and the angle of elevation cu The dimensions of range are L. T.T-2 » »» » )> 9 » LT- a „ L^ T^, and the dimensions of all powers of a are L^ T^. Hence we can draw no inferences as to the manner in which a enters the expres- sion for the range. The dimensions of this expression will depend upon V and g alone. Assume that the range varies as Y"*^". Then L = (LT-i)'»(LT-% __ T m+n nn— m— 2n . giving m + n=ly m+2n = 0, whence m = 2, w = - 1. . Y2 . . Hence the range varies as — when a is given. 38 CHAPTER IV. HYDROSTATICS. 34. The following table of the relative density of water at various temperatures (under atmospheric pressure), iihe density at 4° C. being taken as unity^ is from Eofisetti's results deduced from all the best experiments (Ann. Gb. Phys. X. 461 ; xvii. 370, 1869) :— Temp. Cent. Relative Temp. Relative Temp. Relative Density. Cent. Density. Cent. Density. o •999871 13 •999430 o 35 •99418 1 •999928 14 •999299 40 •99235 2 •999969 15 •999160 45 •99037 3 •999991 16 •999002 50 •98820 4 i-oooooo 17 •998841 55 •98582 5 •999990 18 •998654 60 •98338 6 •999970 19 •998460 65 •98074 7 •999933 20 •998259 70 •97794 8 •999886 22 •997826 76 •97498 9 •999824 24 •997367 80 •97194 10 •999747 26 •996866 85 •96879 11 •999655 28 •996331 90 •96656 12 •999549 30 •995765 100 •95865 35. According to Kupffer's observations, as reduced by Professor W. H. Miller, the absolute density (in grammes per cubic centimetre) at 4° is not 1, but 1 •00001 3. Multiplying the above numbers by this 01£AP. IV. 1 HYDRC )STATICS. 39 factor, we obtain the following table of absolute den- si-ties : — lemp i Density. Temp. Density. Temp. Density. '999884 o 13 •999443 1 o 35 •99469 1 •999941 14 •999312 40 •99236 1 2 •999982 15 •999173 . 45 •99038 3 1000004 16 •999015 1 50 •98821 4 1000013 17 •998854 55 •98583 5 1-000003 18 •998667 1 60' •98339 6 •999983 19 •998473 65 •98075 7 •999946 20 •998272 70 •97795 8 •999899 22 •997839 75 •97499 9 •999887 24 •997380 80 •97195 10 •999760 26 •996879 85 •96880 11 •999668 28 •996344 90 •96557 12 •999562 30 •995778 100 •95866 36. The volume, at temperature t\ of the water which <H5cupies unit volume at 4°, is approximately 1 + A(^ - 4)2 - B{t - 4)2« 4- C{t - 4)3, ^here A = 8^38 X lO-«, B = 3-79 X 10-^ C = 2^24 X 10-8; ^•xid the relative density at temperature ^° is given by the *«tnie formula with the signs of A, B, and C reversed. The rate of expansion at temperature t° is 2A(«-4)-2^6B(«-4)^« x 3C{t-4:)\ In determining the signs of the terms with the frac- tional exponents 2^6 and 1^6, these exponents are to be regarded as odd. 37. The following Table of Densities has been compiled by collating the best authorities, but is only to be taken 40 UNITS AND PHYSICAL CONSTANTS. [chap. aa giving rough approximations. Most of the densities vary between wide limits in different specimens : — Solids. Aluminium, Antimony, .. Bismuth, .... Brass, Copper, Gold, Iron, Lead, Nickel, Platinum, ... Silver, Sodium, Tin, Zinc, Cork Oak, Ebony, Ice, 1 2-6 6-7 9-8 8-4 8-9 19-3 7-8 11-3 8-9 21-5 10-5 •98 7-3 7-1 •24 •7 to 1 •1 to 1-2 •918 >» It Carbon (diamond),., (graphite),., (gas carbon), ,, (wood charcoal), Phosphorus (ordi- nary), ,, (red),... Sulphur (roll), Quartz (rock cry- stal), Sand (dry), Clay, Brick, Basalt, Chalk, 1" Glass (crown), 2' „ (flint) 3" Porcelain, 3 2 I I 5 3 9 6 183 2-2 2^0 2-65 142 1-9 21 3-0 8 to 2-8 5 to 2^7 to 3-5 2-4 Liquids at 0° C. Sea water, 1'026 Alcohol, '8 Chloroform, 1*5 Ether, ^73 Bisulphide of Carbon, . . 1 "29 Glycerine, 1^27 Mercury, 13*596 Sulphuric Acid,... . Nitric Acid, Hydrochloric Acid, Milk, Oil of Turpentine,.. Linseed, it I 85 I^56 I 27 1-03 •87 •94 Mineral, •76to-83 More exactly, the density of mercury at 0° C, as com- pared with water at the temperature of maximum density, under atmospheric pressure, is 13^5956. 38. If a body weighs m grammes in vacuo and m' grammes in water of density unity, the volume of the body is w - m' cubic centims. ; for the mass of the water displaced is m-m/ grammes, and each gramme of this water occupies a cubic centimetre. IV.] HYDROSTATICS. 41 Examples, 1. A glass cylinder, I centims. long, weighs m grammes in vacuo and m! grammes in water of unit density. Find its radius. Solution. Its section is Trr^, and is also — - — ; hence V 2 _ m - m' "^ — vr' 2. Find the capacity at 0° C. of a bulb which holds m grammes of mercury at that temperature. Solution, The specific gravity of mercury at 0° being 13*5956 as compared with water at the temperature of maximum density, it follows that the mass of 1 cubic centim. of mercury is 13-5956 x 1-000013 = 13-5958, say 13-596. Hence the required capacity is cubic lo *o»/0 ^^ntims. 3. Find the total pressure on a surface whose area is A square centims. when its centre of gravity is immei-sed to * depth of h centims. in water of unity density, atmos- pheric pressure being neglected. Ana, Ah grammes weight ; that is, gAh dynes. 4. If mercury of specific gravity 13*596 is substituted fcr water in the preceding question, find the pressure. Ans. 13-596 Ah grammes weight; that is, 13-596 gAh 5. If h be 76, and A be unity in example 4, the answer becomes 1033*3 grammes weight, or 1033-3^^ dynes. For Paris, where g is 980*94, this is 1*0136 xlO« dynes. 42 UNITS AND PHYSICAL CONSTANTS. [chap. Barometric Pressure. 39. The C.G.S. unit of pressure intensity (that is, of pressure per unit area) is the pressure of a dyne per square centim. At the depth of h centims. in a uniform liquid whose density is d [grammes per cubic centim.], the pressure due to the weight of the liquid is ghd dynes per square centim. The pressure-intensity due to the weight of a column of mercuiy at 0° C, 76 centims. high, is found by putting A =76, c?= 13-596, and is 1033%. It is therefore different at different localities. At Paris, where g is 980-94, it is 1-0136 x 10«; that is, rather more than a megadyne* per square centim. To exert a pressure of exactly one megadyne per square centim^^ the height of the column at Paris must be 74*98 centims. At Greenwich, where g is 981-17, the pressure due to 76 centims. of mercury at 0** C. is 1*0138 x 10^; and the height which would give a pressure of 10^ is 74*964 centims.^ or 29*514 inches. Convenience of calculation would be promoted by adopting the pressure of a megadyne per square centim., or 10^ C.G.S. units of pressure-intensity, as the standard atmosphere. The standard now commonly adopted (whether 76 centims. or 30 inches) denotes different pressures at different places, the pressure denoted by it being pro- portional to the value of g. We shall adopt the megadyne per square centim. as our standard atmosphere in the present work. *The prefix mega denotes multiplication by a million. A megadyne is a force of a million dynes. IV.] HYDROSTATICS. 4a Examples, 1. What must be the height of a column of water of unit density to exert a pressure of a megadyno per square centim. at a place where g is 981 ? Ana. ^^^^ = 1019-4 centims. This is 33-445 feet. 2. What is the pressure due to an inch of mercury at 0* C. at a place where g is 981 ? (An inch is 2*54 centims.) Am, 981 X 2-54 x 13*596 = 33878 dynes per square centim. 3. What is the pressure due to a centim. of mercury at 0° C. at the same locality ? Am. 981x13-596 = 13338. 4. What is the pressure due to a kilometre of sea-water of density 1-027, g being 981 1 Ans, 981 X 105 x 1-027 = 1*0075 x 10^ dynes per square centim., or 1-0075 x 10^ megadynes per square centim.; that is, about 100 atmospheres. 5. What is the pressure due to a mile of the same water 1 Ana, 1*6214 x 10^ C.G.S. units, or 162*14 atmospheres [of a megadyne per square centim.]. Density of Air, 40. Regnault found that at Paris, under the pressure of a column of mercury at 0°, of the height of 76 centims., the density of perfectly dry air was -0012932 gramme per cubic centim. The pressure corresponding to this height of the barometer at Paris is 1*0136 x 10^ dynes per square 44 UNITS AND PHYSICAL CONSTANTS. [chap- centiui. Hence, by Boyle's law, we can compute the density of dry air at 0® C. at any given pressure. At a pressure of a megadyne (10® dynes) per square centim. the density will be— -— — ^= '0012759. The density of dry air at 0** C. at any pressure p (dynes per square centim.) is ;?xl-2759xlO-» . . . . W Uocample. Find the density of dry air at 0° C, at Edinburgb, under the pressure of a column of mercury at 0° C, of the height of 76 centims. Here we have p = 981-54 x 76 x 13-596 = 1*0142 x W. Am. Required density = 1-2940 x lO"* = -0012940 gramme per cubic centim. 41. Absolute Densities of Gases, in grammes per cubic centim. y at 0° C, and a pressure q/* 10® dynes per square centim,. Mass of a cubic Volume of a g^ramme centim. in grammes, in cubic centims. 0012759 783-8 0014107 708-9 0012393 806-9 00008837 11316-0 0019509 512-6 0012179 821-1 0007173 1394-1 0030909 323-5 0019433 514-6 0013254 754-5 0026990 370-5 0022990 435-0 0012529 798-1 0007594 1316-8 Air, dry, Oxygen, Nitrogen, Hydrogen, Carbonic Acid, . „ Oxide, Marsh Gas, Chlorine, Protoxide of Nitrogen,.. Binoxide ,, Sulphurous Acid, Cyanogen, defiant Gas, Ammonia, ,] HYDROSTATICS. 45 The numbers in the second column are the reciprocals those in the first. The numbers in the first column are identical with the *X^6<5ific gravities referred to water as unity. Assuming that the densities of gases at given pressure ^-^=id temperature are directly as their atomic weights, we *-^^ve for any gas at zero j9v/A=M316xl0i0m; ^^ denoting its volume in cubic centims., m its mass in immes, p its pressure in dynes per square centim., and its atomic weight referred to that of hydrogen as unity. Height of HoTnogeneous Atmosphere, 42. We have seen that the intensity of pressure at ^pth A, in a fluid of uniform density dy is ghd when the Measure at the upper surface of the fluid is zero. The atmosphere is not a fluid of uniform density ; but is often convenient to have a name to denote a height such that p = ^HD, where p denotes the pressure and the density of the air at a given point. It may be defined as the height of a column of uniform Qidd having the same density as the air at the pointy '^hich would exert a pressure equal to that existing at "^te point. If the pressure be equal to that exerted by a column of ^Xiercury of density 13 '5 9 6 and height A, we have p = ghyi 13-596; .-. HD = A X 13-596, H = ^^ ^ 13-596 If it were possible for the whole body of air above the point to be reduced by vertical compression to the density which the air has at the point, the height from the point 46 UNITS AND PHYSICAL CONSTANTS. [chap. up to the summit of this compressed atmosphere would be equal to H, subject to a small correction for the variation of gravity with height. H is called the height of the homogeneous atmosphere at the point considered. Pressure-height would be a better name. The general formula for it is H=^; ... (5) and this formula will be applicable to any other gas as well as dry air, if we make D denote the density of the gas (in grammes per cubic centim.) at pressure p. If, instead of jp being given directly in dynes per square centim., we have given the height A of a column of liquid of density d which would exert an equal pressure, the formula reduces to H = '^. (6) 43. The value of ^ in formula (5) depends only on the nature of the gas and on the temperature ; hence, for a given gas at a given temperature, H varies inversely •as g only. For dry air at zero we have, by formula (4), ^ 7-8376 X 108 xl = . 9 At Paris, where g is 980*94, we find H = 7-990x105. At Greenwich, where g is 981*17, H = 7-988 xlO^ IV.] HYDROSTATICS. 47 Examples, 1. Find the height of the homogeneous atmosphere at Paris for dry air at 10° C, and also at 100° C. Ans. For given density, p varies as 1 x -00366 t, t de- noting the temperature on the Centigrade scale. Hence we have, at 10° C, H = 1-0366 X 7-99 x 10^ = 8*2825 x 10^ ; and at 100° C, H = 1-366 X 7-99 x 105 = 1-0914 x 10». 2. Find the height of the homogeneous atmosphere for hydrogen at 0°, at a place where g is 981. Here we have K=P- = — = 1-1535x107. gd 981x8-837x10-'^ JDiminution of Density with increase of Height in the Atmosphere. 44. Neglecting the variation of gravity with height, the variation of H as we ascend in the atmosphere would depend only on variation of temperature. In an atmos- phere of uniform temperature H will be the same at all heights. In such an atmosphere, an ascent of 1 centim. will involve a diminution of the pressure (and therefore of the density) by — of itself, since the layer of air which H has been traversed is =_. of the whole mass of superincum- H bent air. The density therefore diminishes by the same fraction of itself for every centim. that we ascend; in other words, the density and pressure diminish in geo- metrical progression as the height increases in arithmetical progression. 48 UNITS AND PHYSICAL CONSTANTS. [chap. Denote height above a fixed level by 05, and pressure by p. Then, in the notation of the differential calculus, , dx dp we nave ._- = - ^ ±1 p and if jt?i, pg ^^ *^® pressures at the heights a^, ajg* ^^ deduce x.-x^ = ll log,?! = H X 2-3026 log^o^^- • • (7) In the barometric determination of heights it is usual to compute H by assuming a temperature which is the arithmetical mean of the temperatures at the two heights. For the latitude of Greenwich formula (7) becomes (Tg - a?! = (1 X -00366 1) 7-988 x 10^ x 2*3026 log^ P2 = (lx -00366 1,839,300 log^i, • • (8) P2 t denoting the mean temperature, and the logarithms being common logarithms. To find the height at which the density would be halved, variations of temperature being neglected, we must put 2 for O in these formulae. The required height will be H P2 log, 2, or, in the latitude of Greenwich, for temperature O"* C, will be 1-8393 X 106 X -30103 = 553700. The value of log, 2, or 2-3026 log^o 2, is 2-3026 X -30103 =-69315. Hence for an atmosphere of any gas at uniform tempera- ture, the height at which the density would be halved is the height of the homogeneous atmosphere for that gas, multiplied by -69315. The gas is assumed to obey Boyle's law. IV.] HYDROSTATICS. 49 Examples. 1. Show that if the pressure of the gas at the lower station and the value of g be given, the height at which the density will be halved varies inversely as the density. 2. At what height, in an atmosphere of hydrogen at 0** C, would the density be halved, g being 981 1 Ana. 7-9954 xlO« 45. The phenomena of capillarity, soap-bubbles, etc., can be reduced to quantitative expression by assuming a tendency in the surface of every liquid to contract. The following table exhibits the intensity of this contractile force for various liquids at the temperature of 20** C. The contractile force diminishes as the temperature in- creases. Sttperficial tensions at 20° C, in dynes per linear centim.^ dediiced from Quincke* s results. Density. Tension of Surface separating the Liquid from Air. Water. Mercury. Water, Mercury, - - - BiBulphide of Carbon, - Chloroform, - Alcohol, OUveOil, - Turpentine, - Petroleum, - Hydrochloric Acid, Solution of Hyposul- ) phite of Soda, - - j 0-9982 13-5432 1-2687 1-4878 •7906 •9136 •8867 -7977 11 11248 81 540 32 1 30-6 25-5 36-9 29-7 31-7 70-1 77-5 418 41-75 29-5 26'-66 11-55 27-8 • • • • • • 418 372-5 399 399 335 250-5 284 377 442-5 The values here given for water and mercury are only i^plicable when special precautions are taken to ensure D 50 UNITS AND PHYSICAL CONSTANTS. [chap. cleanliness and purity. Without such precautions smaller values will be obtained. (Quincke in Wied. Ann,, 1886, page 219.) The following values are from the observations of A. M. Worthington (Proc, Roy, Soc, June 16, 1881), at tempera- tures from 15** to 18° C, for surfaces exposed to air : — ^Surface Tension, In gm. per cm. Water, -072 to -080 Alcohol, -02586 Turpentine, -02818 Olive Oil, 03373 Chloroform, -03025 In dynes per cm. 70-6 to 78-5 25-3 27-6 33-1 29-6 46. Very elaborate measurements of the thicknesses of soap films have been made by Reinold and Riicker (Phil, Trans,, 1881, p. 456; and 1883, p. 651). When so thin as to appear black, the thickness varied from 7 '2 to 14-6 millionths of a millimetre, the mean being 11-7. This is 1-17 X 10~® centimetra The following thicknesses were observed for the colours of the successive orders : — Thickness, cm. First Order— Red, 2-84x10-5 Second Order — Violet, 3-05 Blue, 3-53 Green, 4-09 Yellow, 4-54 Orange, 4-91 Red, 5-22 Third Order— Purple, 5-59 Blue, 5-77 „ 603 Green, 6-66 Thickness, cm. Yellow, 7.10x10-'^ Red, 7-65 Bluish Red, 8-15 >» Fourth Order— Green, 8-41 „ 8-93 Yellow-Green, . . 9-64 Red, 10-52 »» >» »» »> »» ft Fifth Order — Green, 1-119x10-* „ M88 „ Red, 1-260 ,, „ 1-336. >» IV.] HYDROSTATIO=I. 51 Sixth Order — Green, )» Red, >» Sevbnth Obdeu- Green, Thickness, cm. I -410 X 10-* 1-479 „ 1-548 „ 1-627 „ Green, Red,... >» Thickness, cm. 1-787x10-* 1-869 1-936 >> 1-705 Eighth Order- Green, Red, »» 2004 „ 2115 „ ») 46a. Depression of the barometrical column due to capillarity, according to Pouillet : — Internal Internal Internal Diameter Depression. Diameter Depression. Diameter Depression. of tube. of tube. of tube. inm. mm. mm. nun. mm. mm. 2 4-579 8-5 •604 15 •127 2-6 3-595 9 •534 15-5 -112 3 2-902 9-5 •473 16 -099 3-5 2-415 10 •419 16-0 •087 4 2-053 10-5 •372 17 -077 4-5 1-752 11 •330 17 5 •068 ,5 1-507 11-5 •293 18 •060 5-6 1-306 12 •260 18^5 -053 6 1-136 12-5 -230 19 •047 6-5 •995 13 •204 195 •041 7 -877 13-5 -181 20 -036 7-5 -775 14 •161 205 -032 8 •684 14-5 -143 21 -028 52 CHAPTER V. STRESS, STRAIN, AND RESILIENCE. 47. In the nomenclature introduced hy Kankine, and adopted hy Thomson and Taifc, any change in the shape or size of a body is called a strain, and an action of force tending to produce a strain is called a stress. We shall always suppose strains to be small ; that is, we shall sup- pose the ratio of the initial to the final length of every line in the strained body to be nearly a ratio of equality. 48. A strain changes every small spherical portion of che body into an ellipsoid ; and the strain is said to be homogeneous when equal spherical portions in all parts of the body are changed into equal ellipsoids with their corresponding axes equal and parallel. When the strain consists in change of volume, unaccompanied by change of shape, the ellipsoids are spheres. When strain is not homogeneous, but varies continu- ously from point to point, the strain at any point is defined by attending to the change which takes place in a very small sphere or cube having the point at its centre, so small that the strain throughout it may be regarded as homogeneous. In what follows we shall suppose strain to be homogeneous, unless the contrary is expressed. CHAP, v.] STRESS, STRAIN, AND RESILIENCE. 53 49. The axes of a strain are the three directions in the body, at right angles to each other, which coincide with the directions of the axes of the ellipsoids. Lines drawn in the body in these three directions will remain at right angles to each other when the body is restored to its unstrained condition. A cube with its edges parallel to the axes will be altered by the strain into a rectangular parallelepiped. Any other cube will be changed into an oblique parallele- piped. When the axes have the same directions in space after sua before the strain, the strain is said to be unaccompanied- by rotation. When such parallelism does not exist, the strain is accompanied by rotation, namely, by the rotation which is necessary for bringing the axes from their initial to theii' final position. The numbers which specify a strain are mere ratios, and are therefore independent of units. 50. When a body is under the action of forces which strain it, or tend to strain it; if we consider any plane section of the body, the portions of the body which it separates are pushing each other, pulling each other, or exerting some kind of force upon each other, across the section, and the mutual forces so exerted are equal and opposite. The specification of a stress must include a specification of these forces for all sections, and a body is said to be homogeneously stressed when these forces are the same in direction and intensity for all parallel sec- tions. We shall suppose stress to be homogeneous, in what follows, unless the contrary is expressed. 51. When the force-action across a section consists of 54 UNITS AND PHYSICAL CONSTANTS. [chap. a simple pull or push normal to the section, the direction of this simple pull or push (in other words, the normal to the section) is called an axis of the stress. A stress (like a strain) has always three axes, which are at right angles to one another. The mutual forces across a section not perpendicular to one of the three axes are in general partly normal and partly tangential — one side of the sec- tion is tending to slide past the other. The force per unit area which acts across any section is called the intensity of the stress on this section, or simply the stress on this section. The dimensions of " force per unit area," or — are - —,, which we shall therefore call area LT2' the dimensions 0/ stress. 52. The relation between the stress acting upon a body and the straih produced depends upon the resilience of the body, which requires in general 21 numbers for its complete specification. When the body has exactly the same properties in all directions, 2 numbers are sufficient. These specifying numbers are usually called coefficients of elasticity; but the word elasticity is used in so many senses that we prefer to call them coefficients 0/ resilierice. A coefficient of resilience expresses the quotient of a stress (of a given kind) by the strain (of a given kind) which it produces. A highly resilient body is a body which has large coefficients of resilience. Steel is an example of a body with large, and cork of a body with small, coefficients of resilience. In all cases (for solid bodies) equal and opposite strains (supposed small) require for their production equal and opposite stresses. v.] STRESS, STRAIN, AND RESILIENCE. 55 53. The coefficients of resilience most frequently re- ferred to are the three following : — (1) Resilience of volume, or resistence to hydrostatic compression. If V be the original and Y -v the strained volume, — is called the compression, and when the body is subjected to uniform normal pressure P per unit area over its whole surface, the quotient of P by the compres- sion is the resilience of volume. This is the only kind of resilience possessed by liquids and gases. (2) Young's modulus, or the longitudinal resilience of a body which is perfectly free to expand or contract laterally. In general, longitudinal extension produces lateral contraction, and longitudinal compression produces lateral extension. Let the unstrained length be L and the strained length Ij±1, then - is taken as the measure of the longitudinal extension or compression. The stress on a cross section (that is, on a section to which the stress is normal) is called the longitudinal stress, and Young's modulus is the quotient of the longitudinal stress by the longitudinal extension or compression. If a wire of cross section A sq. cm. is stretched with a force of F dynes, and its length is thus altered from L to L + /, the value ■p T of Young's modulus for the wire is - . y. (3) " Simple rigidity " or resistance to shearing. This requires a more detailed explanation. 54'. A -shear may be defined as a strain by which a sphere of radius unity is converted into an ellipsoid of semiaxes 1, 1+6, I -e; in other words, it consists of an 56 UNITS AND PHYSICAL CONSTANTS. [chap. extension in one direction combined with an eqnal com- pression in a perpendicular direction. 55. A unit square (Fig. 1) whose diagonals coincide with these directions is altered by the strain into a rhombus whose diagonals are (1 + e) J2 and (1 - e) ,J% and whose area, being half the product of the diagonals, is 1-6^, or, to the first order of small quantities, is 1, the same as the area of the original square. The length of a side of the rhombus, being the square root of lihe Fig.i rt^.z sum of the squares of the semi-diagonals, is found to be Vl + e^ or 1 + ^\ and is therefore, to the first order of small quantities, equal to a side of the original square. 56. To find the magnitude of the small angle which a side of the rhombus makes with the corresponding side of the square, we may proceed as follows :— Let ach (Fig. 2) be an enlarged representation of one of the small tri- angles in. Fig. 1 . Then we have ab = ^, cb = ^e J2 = Ji' TT angle cha = -. Hence the length of the perpendicular cd IT is ch sin^ = -. - — ^ = - : and since ad is ultimately equal to a6, we have, to the first order of small quan- tities. v.] STRESS, STRAIN, AND RESILIENCE. 57 , , cd ie angle cab = -3= ^ = e. ad ^ TT The semi-angles of the rhombns are therefore - ± e, IT and the angles of the rhombus are - ± 2e ; in other words, each angle of the square has been altered by the amount 2e, This qiuintity 2e is adopted as the measure of the shear, 57. To find the perpendicular distance between oppo- site sides of the rhombus, we have to multiply a side by the cosine of 2e, which, to the first order of small quan- tities, is 1. Hence the perpendicular distance between opposite sides of the square is not altered by the shear, and the relative movement of these sides is represented ^^ff ^ rig. 4 by supposing one of them to remain fixed, while the other slides in the direction of its own length through a <iistance of 2e, as shown in Fig. 3 or Fig. 4. Fig. 3, in fact, represents a shear combined with right-handed rota- tion, and Fig. 4 a shear combined with left-handed rota- tion, as appears by comparing these figures with Fig. 1, which represents shear without rotation. 58. The square and rhombus in these three figures may be regarded as sections of a prism whose edges are per- pendicular to the plane of the paper, and figures 3 and 4 68 UNITS AND PHYSICAL CONSTANTS. [chap. show that (neglecting rotation) a shear consists in the relative sliding of parallel planes without change of dis- tance, the amount of this sliding being proportional to the distance, and being in fact equal to the product of the distance by the numerical measure of the shear. A good illustration of a shear is obtained by taking a book, and making its leaves slide one upon another. It may be well to remark, by way of caution, that the selection of the planes is not arbitrary as far as direction is concerned. The only planes which are affected in the manner here described are the two sets of planes which make angles of 45° with the axes of the shear (these axes being identical with the diagonals in Fig. 1). 59. Having thus defined and explained the term " shear," which it will be observed denotes a particular species of strain, we now proceed to define a shearing stress. A shearing stress may be defined as the combination of two longitudinal stresses at right angles to each other, these stresses being opposite in sign and equal in magni- tude ; in other words, it consists of a pull in one direction combined with an equal thrust in a perpendicular direction. 60. Let P denote the intensity of each of these longitudinal stresses; we shall proceed to cal- culate the stress upon a plane in- I ■*' clined at 45° to the planes of these . stresses. Consider a unit cube so " ' taken that the pull is perpendicular to two of its faces, AB and DC (Fig. 5), and the thrust v.] STRESS, STRAIN, AND RESILIENCE. 59 is perpendicular to two other faces, AD, BC. The forces which hold the half-cube ABC in equilibrium are — (1) An outward force P, uniformly distributed over the face AB, and having for its resultant a single force P acting outward applied at the middle point of AB. (2) An inward force P, having for its resultant a single force P acting inwards at the middle point of BC. (3) A force applied to the face AC. To determine this third force, observe that the other two forces meet in a point, namely, the middle point of AC, that their components perpendicular to AC destroy one another, and that their components along AC, or p rather along CA, have each the magnitude -- - ; hence their resultant is a force P ^2, tending from C towards A. The force (3) must be equal and opposite to this. Hence each of the two half-cubes ABC, ADC exerts upon the other a force P ^2, which is tangential to their plane of separation. The stress upon the diagonal plane AC is therefore a purely tangential stress. To compute its intensity we must divide its amount P ^2 by the area of the plane, which is ^2, and we obtain the quotient P. Similar reasoning applies to the other diagonal plane BD. P is taken as the measure of the shearing stress. The above discussion shows that it may be defined as the intensity of tlie stress either on the planes of purely normal stress^ or on the planes of purely tangential stress, 61. A shearing stress, if applied to a body which has the same properties in all directions (an isotrojnc body), produces a simple shear with the same axes as the stress ; for the extension in the direction of the pull will be equal to the compression in the direction of the thrust ; and in 60 UNITS AND PHYSICAL CONSTANTS. [chap. the third direction, which is perpendicular to both of these, there is neither extension nor contraction, since the transverse contraction dae to the pull is equal to the transverse extension due to the thrust. A shearing stress applied to a body which has not the same properties in all directions produces in general a shear with the same axes as the stress, combined with some other distortion. In both cases, the quotient of the shearing stress by the shear produced is called the resistance to shea/ring. In the case of an isotropic body, it is also called the simple rigidity, 62. The following values of the resilience of liquids under compression are reduced from those given in Jamin, ^*Oours de Physique," 2nd edition, torn, i pp. 168 and 169 :— Compression for Temp Coeflaclent of one Atxnosphere Cent. Resilience. (m^adyne per square centim.) Mercury, - 00 3-436x1011 1 2-91 X 10-« Water, 0-0 2 02 xlO^o 4-96 X 10-» ,. - - 1-6 1-97 „ 508 „ .J - - 41 2-03 „ 4-92 „ »» 10-8 2-11 „ 4-73 ,, ,, 13-4 2-13 „ 4-70 „ , . 18-0 2-20 ,, 4-65 „ )> 25 2-22 „ 4-50 „ »» 34-5 2-24 „ 4-47 „ j» " ■ 43 2-29 „ 4*36 „ »j " ~ 53-0 2-30 „ 4-35 „ r 0*61 9-2 X 10» 109x10-* Ether, - - 00 V 7-8 „ 1 -29 „ U4-0j 7-2 „ 1-38 „ Alcohol, 7-3 { \ 13-1 i 1-22 xlO^o 112 „ 8-17 „ 8-91 X 10-» Sea Water, - 17-5 2-33 „ 4-30 „ ^.] STRESS, STRAIN, AND RESILIENCE. 61 63. The following are reduced from the results ob- tained by Amaury and Descamps, " Comptes Bendus/' torn. Ixviii. p. 1564 (1869), and are probably more accurate than the foregoing, especially in the case of mercury : — Coefficient of Resilience. Compression for one megadyne per square centim. Distilled Water, Alcohol, - / „ ... Ether, »> - - - / Bisulphide of Carbon, Mercury, - o 15 15 14 14 15 2-22 X 1010 1-21 „ 111 „ 9-30x109 7-92 „ 1-60x1010 5-42 X 1011 4-51 X 10-5 8-24 „ 8-99 „ 1-08x10-4 1*26 „ 6-26x10-5 1-84x10-6 64. The following values of the coefficients of resilience 'or solids are reduced from those given in my own papers o the Royal Society (see « Phil. Trans.," Dec. 5th, 1867, ^ 369), by employing the value of g at the place of ob- ^rvation^ namely, 981*4. Young's Modulus. Simple Rigi(fity. Resilience of Volume. Density CHass, flmt. 6-03 xlOii 2-40 X 1011 4-15 xlOii 2-942 -Another specimen 5-74 „ 2-35 „ 3-47 „ 2-935 ^rass, drawn, - 1-075x101^ 3-66 „ 8-471 «teel,- 2-139 „ 8-19 ,, 1-841x101^ 7-849 Xx)n, wrought, - 1-963 „ 7-69 „ 1-456 „ 7-677 „ cast,- 1-349 „ 5-32 „ 9-64 xlOii 7-236 Copper, 1-234 „ 4-47 „ 1-684x101*^ 8-843 65. The resilience of volume was not directly observed, ^ut was calculated from the values of " Young's modulus " ^ud "simple rigidity,'* by a formula which is strictly true 62 UNITS AND PHYSICAL CONSTANTS. [chap. for bodies which have the same properties in all direc- tions. The contraction of diameter in lateral directions for a body which is stretched by purely longitudinal stress was also calculated by a formula to which the same remark applies. The ratio of this lateral contraction to the longitudinal extension is called " Poisson's ratio," and the following were its vs^lues as thus calculated for the six bodies experimented on : — Glass, flint, '258 Another specimen, '229 Brass, drawn, '469 (?) Steel, -310 Iron, wrought, '275 ,, cast, '267 Copper, -378 Kirchhoff has found for steel the value '294, and Clerk Maxwell has found for iron '267. Comu ("Oomptes Rendus," August 2, 1869) has found for different speci- mens of glass the values -225, '226, '224, -257, '236, -243, •250, giving a mean of '237, and maintains (with many other continental savants) that for all isotropic solids (that is, solids having the same properties in all direc- tions) the true value is ^. 66. The following values of Poisson's ratio have been found by Mr. A. Mallock (" Proc. Roy. Soc," June 19, 1879) :— Steel, Brass, Copper, Lead, Zinc (rolled), ,, (cast),... Ebonite, •253 •325 •348 •375 •180 •230 •389 Ivory, '50 India Rubber, '50 Paraffin, '50 Plaster of Paris,. . . . '181 Cardboard, '2 Cork, -00 Boxwood,... Beechwood, , White Pine, Radial due to LongitudinaL •42 •53 •486 Longitudinal due to Radial. •406 •408 •372 In Cross Section. •227 v.] STRESS, STRAIN, AND RESILIENCE. 63 The heading "Radial due to Longitudinal" means that the applied force is longitudinal (that is, parallel to the length of the tree) and that the contraction along a radius of the tree is compared with the longitudinal extension. 67. The following are reduced from Sir W. Thomson's results ("Proc. Roy. Soc," May, 1865), the value of g being 981-4:— Simple Rigidity. Brass, three specimens, 4*03 3*48 3*44) j^u Copper, two specimens, 4*40 4*40 { Other specimens of copper in abnormal states gave i-esults ranging from 3*86 x 10^^ to 4*64 x 10^^ The following are reduced from Wertheim's results (" Ann. de Ohim.," ser. 3, tom. xxiii.), g being taken as 981:— Different Specimens of Glass {Crystal), Young's Modulus, 3*41 to 4*34, mean 3*96 ^ Simple Rigidity, 1 -26 to 1 '66 „ 1 '48 }- x 10" Volume Resilience 3*50 to 4*39 ,, 3 '89 J Different Specimens of Brass. ] Young's Modulus, .... 9 '48 to 10 44, mean 9 '86 Simple Rigidity, 3*53 to 3*90 „ 3*67 }- x lO^i Volume Resilience,.. 1002 to 10-85 „ 10-43 68. Savart's experiments on the torsion of brass wire (« Ann. de Chim.," 1829) lead to the value 3-61 x 10" for simple rigidity. Kupffer's values of Young's modulus for nine different specimens of brass range from 7*96 x 10" to 11*4 x 10^^, the value generally increasing with the density. For a specimen, of density 8*4465, the value was 10-58 X 10". 64 UNITS AND PHYSICAL CONSTANTS, [chap. v. For a specimen, of density 8*4930, the value was 11-2x10". The values of Young's modulus found by the same experi- menter for steel, range from 20*2 x 10" to 21*4 x 10". 69. The following are reduced from Kankine's '' Rules and Tables," pp. 195 and 196, the mean value being adopted where different values are given : — Tenacity. Young's Modulus. Steel Bars, 7'93xlO» 2*45 x lO^^ IronWire 6'86 „ 1745 „ Copper Wire, 4*14 „ 1-172 „ BrassWire, 338 „ 9-81 x lO" Lead, Sheet, 2-28xl0« 5*0 x IQio Tin, Cast, 3*17 „ Zinc, 5-17 ,, Ash, 1-I72xl0» 1-10 xlQii Spruce, 8-55 x lO^ I'lO Oak, l-026xl0» 1-02 99 »» Glass, 6-48xl0« 6*52 x IQii Brick and Cement,.. 2*0 x 10" The tenacity of a substance may be defined as the greatest longitudinal stress that it can bear without tear- ing asunder. The quotient of the tenacity by Young's modulus will therefore be the greatest longitudinal exten- sion that the substance can bear. 65 CHAPTER VI. ASTRONOMY. Size and Figure of the Earth. 70. According to the latest determination, as pub- liahed by Capt. Clarke in the " Philosophical Magazine " £or August, 1878, the semiaxes of the ellipsoid which most nearly agrees with the actual earth are, in feet, a = 20926629, b - 20925105, c = 20854477, -which, reduced to centimetres, are a = 6-37839 x lO^, b = 6-37792 x lO^, c = 6-35639 x lO^, giving a mean radius of 6*3709 x 10^, and a volume of 1-0832 x 1027 cubic centims. The ellipticities of the two principal meridians are 1 and 1 289-5 296-8 The longitude of the greatest axis is S"" 15' W The mean ieiigth of a quadrant of the meridian is 1*00074 x 10^. The length of a minute of latitude is approximately 1 85200 - 940 cos. 2 lat. of middle of arc. The mass of the earth, assuming Baily's value 5*67 for mean density, is 6*14 x 10^^ grammes. £ r, r ©"-' I 66 UNITS AND PHYSICAL CONSTANTS. [chap. Day and Year. Sidereal day, 86164 mean solar seconds. Sidereal year, 31,558,150 „ Tropical year, 31,556,929 „ 2x 1 Angular velocity of earth's rotation, = ^^5=^0. 861o4 13713 Velocity of earth in orbit, about 2960600 „ *^*Si iXtiot T.'*!r '"!} 3-3908 dynes per g«nune. Attraction in Astronomy, 71. The mass of the moon is the product of the earth's mass by -011364, and is therefore to be taken as 6-98 X 102^ grammes, the doubtful element being tiie earth's mean density, which we take as 5*67. The mean distance of the centres of gravity of the earth and moon is 60*2734 equatorial radii of the earth —that is, 3-8439 x lO^^ centims. The mean distance of the sun from the earth is about 1-487 X 10^^ centims., or 92-39 million miles, correspond- ing to a parallax of 8"-848.* The intensity of centrifugal force due to the earth'i "T ) noting the mean distance, and T the length of the sidereaEI year, expressed in seconds. This is equal to the accelera- tion due to the sun's attraction at this distance. Puttin for r and T their values, 1-487 x lO^s and 3-1558 x 10 we have 1 -r^ 1 r = -5894. * This value of the mean solar parallax was determined by Prt^ -^ feesor Newcomb, and was adopted in the '* Nautical Almanac lor 1882. (See Art. 86 for a later determination.) VI.] ASTRONOMY. 67 This is about , ^^^ of the value of g at the earth's 1660 ^ surface. The intensity of the earth's attraction at the mean dis- tance of the moon is about -^?1- or -2701. (60-27)2 This is less than the intensity of the sun's attraction upon the earth and moon, which is *5894 as just found. Hence the moon's path is always concave towards the sun. 72. The mutual attractive force F between two masses m and m', at distance I, is where is a constant. To determine its value, consider the case of a gramme at the earth's surface, attracted by the earth. Then we have F = 981, m = l, m' = 6-14 x 1027, ; = 6-37 x lO^; whence we find C = ^= -^ 108 1.543x107* TTo find the mass m which, at the distance of 1 centim. ^rom an equal mass, would attract it with a force of 1 <iyne, we have 1 = Qrn^ ; 'vrhence m = . - = 3928 grammes. 73. To find the acceleration a produced at the distance of I centims. by the attraction of a mass of m grammes, ^e have a = — - = C— , m l^ where C has the value 6*48 x 10"® as above. 68 UNITS AND PHYSICAL CONSTANTS. [chap. To find the dimensions of C we have C = — , where the m dimensions of a are LT~^. The dimensions of C are therefore 1^2 3^-1 LT-2 . i-hat is, L» M-^ T'l 74. The equation a = C^ shows that when a = 1 and ^=1, 971 must equal — ; that is to say, the mass which produces unit acceleration at the distance of 1 centimetre is 1*543 X 10^ grammes. If this were taken as the unit of mass, the centimetre and second being retained as the units of length and time, the acceleration produced by the attraction of any mass at any distance would be simply the quotient of the mass by the square of the distance. It is thus theoretically possible to base a general system of units upon two fundamental units alone ; one of the three fundamental units which we have hitherto employed being eliminated by means of the equation mass = acceleration x (distance)^, which gives for the dimensions of M the expression U T-l Such a system would be eminently convenient in astro- nomy, but could not be applied with accuracy to ordinary terrestrial purposes, because we can only roughly compare the earth's mass with the masses which we weigh in our balances. 75. The mass of the earth on this system is the product of the acceleration due to gravity at the earth's surface, and the square of the earth's radius. This product is 981 X (6-37 X 108)2 = 3-98 X IO20, \ VI.] ASTRONOMY. 69 and is independent of determinations of the earth's density. The new unit of force will be the force which, acting iipon the new unit of mass, produces unit acceleration. It will therefore be equal to 1*543 x 10^ dynes; and its dimensions will be mass X acceleration = (acceleration)^ x (distance)^ = L* T-*. 76. If we adopt a new unit of length equal to I oentims., and a new unit of time equal to t seconds, while we define the unit mass as that which produces unit acceleration at unit distance, the unit mass will be Pt'^ X 1*543 X 10^ grammes. If we make I the wave-length of the line F in vacuo, say, 4-86 x 10"*, and t the period of vibration of the same ray, so that is the velocity of light in vacuo, say, 3 X 1010, / 7 the value of P t"^ or I i^'- 4-374 X 10i«, and the unit mass will be the product of this quantity into 1*543 x 10^ grammea This product is 6*75 x 10^3 grammes. The mass of the earth in terms of this unit is 3*98 x 1020 ^ (4*374 x 10i«) = 9100, and is independent of determinations of the earth's density. CHAPTER VII. VELOCITY OF SOUND. 77. The propagation of sound through any medium is due to the elasticity or resilience of the medium ; and the general formula for the velocity of propagation s is 8 where D denotes the density of the medium, and E the coefficient of resilience. 78. For air, or any gas, we are to understand by E the quotient increment of pressure corresponding compression ' that is to say, if P, P + p be the initial and final pres- sures, and V, V - V the initial and final volumes, p and v being small in comparison with P and V, we have V V V If the compression took place at constant temperature, we should have | = |.andE = P. But in the propagation of sound, the compression is efiected so rapidly that there is not time for any sensible part of the heat of compression to escape, and we have CHAP. VII.] VELOCITY OF SOUND. 71 where y = 1'41 for dry air, oxygen, nitrogen, or hydrogen. The value of — for dry air at f Cent, (see p. 46) is (1 + -003660 ^ 7-838 + lO^. Hence the velocity of sound through dry air is «= W >/l-41x(l + '00366<)x 7-838 = 33240 VI + .00366«; or approximately, for atmospheric temperatures, 8 = 33240 + 60<. 79. In the case of any liquid, E denotes the resilience of volume.* For water at 8***1 C. (the temperature of the Lake of G^eneva in Colladon's experiment) we have E = 2-08 X 1010, D = 1 sensibly ; V 1= VE = 144000, the velocity as determined by CoUadon was 143600. 80. For the propagation of sound along a solid, in the form of a thin rod, wire, or pipe, which is free to expand or contract laterally, E must be taken as denoting Young's modulus of elasticity.* The values of E and D will be different for different specimens of the same material. Employing the values given in the Table (§ 64), we have * Strictly speaking, E should be taken as denoting the resili- ence for sudden applications of stress — so sudden that there is not tim^e for changes of temperature produced by the stress to be sensibly diminished by conduction. This remark applies to both §§ 79 and 80. For the amount of these changes of temperature, see a later section under Heat. 72 UNITS AND PHYSICAL CONSTANTS. [chap. Values of E. Values of D. Values of ^5, or velocity. Glass, first specimen, „ secona specimen, Brass, Steel, Iron, wrought, - „ cast, - Copper, , - 6-03 xlO" 6-74 1-075x1112 2-139 „ 1-963 „ 1-349 „ 1-234 „ 2-942 2-935 8-471 7-849 7-677 7-235 8-843 4-63 X W 4-42 „ 3-66 „ 5-22 „ 6-06 „ 4-32 „ 3-74 „ 81. If the density of a specimen of red pine be '5, and its modulus of longitudinal elasticity be 1-6 x 10^ pounds per square inch at a place where g is 981, compute the velocity of sound in the longitudinal direction. By the table of stress, page 4, a pound per square inch {g being 981) is 6-9 x 10* dynes per square centim. Hence we have for the required velocity /E^ / l-6x 100x6-9x1 0* centims. per second. 82. The following numbers, multiplied by 10^, are the velocities of sound through the principal metals, as determined by Wertheim : — = 4-7 X 10^ At 20' C. At 100" C. At 200* C. Lead, 1-23 1-20 • • • Gold, 1-74 1-72 1-73 Silver, 2-61 2-64 2-48 Copper, - 3-56 3-29 2-95 Platmum, 2-69 2-57 2-46 Iron, 6-13 5-30 4-72 Iron Wire (ordinary), 4-92 510 • • • Cast Steel, 4-99 4-92 4-79 Steel Wire (English), 4-71 5-24 5-00 » " " 4-88 5 01 • • • vn.] VELOCITY OF SOUND. 73 The following velocities in wood are from the observa- tions of Wertheim and Chevandier, " Comptes Rendus," 1846, pp. 667 and 668 :— Along Fibres. Radial Direction. Tangential Direction. Pine, 3*32 xO« 2-83 X W l-69xl0« Beech, - 3*34 ,, 3-67 „ 2-83 „ Witch-Ehn, - 3-92 „ 3-41 „ 2-39 „ Birch, - 4-42 „ 2-14 „ 303 „ Fir, - - - 4-64,, 2-67 „ 1-57 „ Acacia, - 4-71 „ AspeD, - 6-08 „ Musical Strings. 83. Let M denote the mass of a string per unit length, F „ stretching force, L „ length of the vibrating portion ; then the velocity with which pulses travel along the string is "=Vm' and the number of vibrations made per second is V Bxample. For the four strings of a violin the values of M in ;grammes per centimetre of length are •00416, -00669, -0106, -0266. T!he values of n are 660, 440, 293J, 195|; 74 UNITS AND PHYSICAL CONSTANTS. [oHAP.vn, and the common value of L is 33 centims. Hence the values of v or 2Ln are 43560, 29040, 19360, 12910 centims. per second ; and the values of F or Mt;^, in dynes, are 7-89 X 10«, 5-64 X 10«, 3-97 x 10«, 4-43 x 10». Faintest Audible Sound. 84. Lord Rayleigh (" Proc. R. S.," 1877, vol. xxvi p. 248), from observing the greatest distance at which a whistle giving about 2730 vibrations per second, and blown by water-power, was audible without effort in the middle of a fine still winter's day, calculates that the maximum velocity of the vibrating particles of air at this distance from the source was *0014 centims. per second, and that the amplitude was 8*1 x 10"^ centims., the calculation being made on the supposition that the sound spreads- uniformly in hemispherical waves, and no deduction being made for dissipation, nor for waste energy in blowing. 75 CHAPTER VIII. LIGHT. 85. All kinds of light are believed to have the same velocity in vacuo. The velocity of light of given re- frangibility in any medium is - of .its velocity in vacuo, A* /x denoting the absolute index of refraction of that medium for light of the given refrangibility. Light of given refrangibility is light of given wave- frequency. Its wave-length in any medium is the quotient of its velocity in that medium by its wave- frequency. If n denote the wave-frequency (that is to say, the number of waves which traverse a given point in a second), the wave-length in any medium will be — of the velocity in vacuo. The absolute index of refraction for ordinary air is about 1*00029. More accurate statements of its value will be found in Arts. 94-96. 86. The best determination of the velocity of light is that made by Professor Newcomb at Washington in 1882 (" Astron. Papers of Amer. Ephem.," vol. ii. parts iii and iv. 1885). The method employed was that of the revolving mirror, the distance between the revolving and 76 UNITS AND PHYSICAL CONSTANTS. [chap. the fixed mirror being ia one portion of the observations 2550 metres, and in the remaining portion 3720 metres. The resulting velocity in vacuo is 2-99860 X 10^0 centims. per sea The following summary of results is from Professor Newcomb's paper, page 202 : — km. per. sec 299910 299853 I 299860 299810 298000 298500 300400 299990 301382 Michelson, at Naval Academy, in 1879, Michelson, at Cleveland, 1882, Newcomb, at Washington, 1882, using only results supposed to be nearly free from constant errors, Newcomb, including all determinations, Foucault, at Paris, in 1862, Comu, at Paris, in 1874, Comu, at Paris, in 1878, This last result as discussed by Listing, Young and Forbes, 1880-81, Professor Newcomb remarks (page 203) that the value 299860 km. per sec. for the velocity of light, combined with Clark's value 6378 '2 km. for the earth's equatorial radius, and Nyren's value 20"'492 for the constant of aberration, gives for the solar pai*allax the value 8" '7 94. 87. The following are the wave-lengths adopted by Angstrom for the principal Fraunhofer lines in air at 760 millims. pressure (at XJpsal) and 16* C. : — Centims. A 7-604 xlO-5 B 6-867 C 6-56201 Mean of Hues D .5-89212 E 5-26913 F 4-86072 G 4-30725 Hi 3-96801 Hs 3-93300 vm.] LIGHT. 77 These nambers will be approximately converted into the corresponding wave-lengths in vacuo by multiplying them by 1-00029. 88. Assuming 3 x 10^^ to be the velocity of light in air, and neglecting the difference of velocity between the more and less refrangible rays, we obtain the follow- ing frequencies by dividing the common velocity by Angstrom's values of the wave-lengths : — Vibrations per Second. A 3045x101* B 4-369 „ C 4-672 „ D 5-092 „ E 5-693 „ F 6-172 „ G 6-965 „ Hi 7-560 „ H, 7-628 „ According to Langley (" Com. Hen./' Jan., 1886), the solar spectrum extends beyond the red as far as wave- length 27 X 10"^ and the radiation from terrestrial bodies at temperatures below 100° extends as far as wave-length 150 X 10"^ The frequencies corresponding to these two wave-lengths are 1*1 x 10^* and 2 x 10^^. INDICES OF REFRACTION OF SOLIDS. 89. Dr. Hopkinson ("Proc. R. S.," June 14, 1877) has determined the indices of refraction of the principal varieties of optical glass made by Messrs. Chance, for the fixed lines A, B, C, D, E, 6, F, (G), G, h, H^. By D is to be understood the more refrangible of the pair of sodium lines ; by b the most refrangible of the group of magnesium lines ; by (G) the hydrogen line near G. 78 UNITS AND PHYSICAL CONSTANTS. [chap. In connection with the results of observation, he employs the empirical formula fjL-l =a{l -hbx^l+cx)}, where a; is a numerical name for the definite ray of which /A is the refractive index. In assigning the valae of x, four glasses— hard crown, soft crown, light flint, and dense flint — were selected on account of the good accord of their results ; and the mean of their indices for any given ray being denoted by /I, the value assigned to x for this ray is /a - fip where /Ip denotes the value of /x for the line F. The value of /a as a function of A., the wave-length in 10"* centimetres, was found to be approximately fx = 1-538414 + 0-0067669^2 - 0-00017341 A A + 0-000023-1. A^ The following were the results obtained for the diflerent specimens of glass examined : — Hard Crown, 1st specimen, density 2*48575. a=0-523145, 5 = 1-3077, c= -2-33. Means of observed values of fi. A 1-611755 E 1-520324 G 1-528348 B 1-513624 b 1-520962 h 1-530904 C 1-514571; D 1-517116; F 1-523145; (G) 1*527996; Hi 1-532789. Soft Crown, density 2-55035. a =0-5209904, 6=1*4034, c=-l-58. Means of observed values of fi. A 1-508956 E 1*518017 G 1*626692 B 1-510918 h 1-518678 h 1-529360 C 1-511910; D 1*514580; F 1*620994; (G) 1*626208; Hj 1*531415. vin.] LIGHT. 79 Extra Light Flint Glass, density 2*86636. a=0-549123, 6=1-7064, c=-0198. Means of observed values of fi» A 1-634067 D 1-541022 P 1-549125 h 1*559992 B 1 -536450 E 1*545295 (G) 1-555870 Hi 1-562760. C 1*537682 b 1*546169 G 1*556375 LigM Flint Glass, density 3*20609. a=0*583887, 6 = 1*9605, c=0-53. Means of observed values of fi. B 1*568558 K 1-579227 <G) 1*592184 H, 1-600717. C 1*570007 6 1*580273 G 1*592825 D 1*574013 F 1*583881 k 1*597332 Dense Flint, density 3*65865. a = 0*634744, 6 = 2-2694, c = l*48. Means of observed values of /<. B 1*615704 £ 1-628882 (G) 1*645268 Hi 1*656229. C 1*617477 6 1*630208 G 1*646071 D 1*622411 F 1*634748 h 1*651830 Extra Dense Flint, density 3 '88947. a=0*664226, 6=2*4446, c=l*87. Means of observed values of fi. A 1-639143 D 1-650374 F 1-664246 A 1-683575 B 1*642894 E 1-657631 (G)* 1*676090 Hi 1*688590. 1*644871 6 1*659108 G 1*677020 80 UNITS AND PHYSICAL CONSTANTS. [chap. Double Extra Dense FlnU, density 4*42162. a=0-727237, ^=2*7690, c=2-70. Means of observed values of /jl. A 1-696531; D 1-710224; F 1-727257; h 1-751485. B 1-701080 E 1-719081 (G) 1-742058 C 1-703485 b 1-720908 G 1-743210 90. The following indices of rock salt, sylvin, and alam for the chief Fraunhofer lines are from Stefan's observer tions : — A B D E F G H Bock Salt at 17' 0. 1-53663 •53918 •54050 •54418 -54901 •55324 -56129 -56823 Sylvin at 20* C. 1-48377 •48597 •48713 •49031 •49455 •49830 •50542 •51061 Alum at 2V C. 1-46057 •45262 -45359 •45601 •45892 •46140 •46563 •46907 91. Indices of other singly refracting solids — Index of Kind of Refraction. Light. Diamond, 2-470 D Fluor-spar, 1*4339 D Amber, 1*532 D Rosin, 1545 Red Copal, 1*528 Red Gum Arabic, 1*480 Red Peru Balsam 1 '593 D Canada Balsam,. 1-528 Red Observer. Schrauf. Stefan. Kohlrausch. Jamin. it >i Baden Powell. Wollaston. Hffhct of Temperature, According to Stefan, the index of refraction of glass increases by about -000002 for each degree Cent, of Tm.] LIGHT. 81 increase of temperature, and the index of rock salt diminiahea hj aboat *000 037 for each degree of increase of temperature. 92. Doubly refracting crystals : — Uniaacal Crystals, rw.»n.«« Bxtraordi- Kind "JJ^*J7 nary of Temp. Obaerver. luaex. Index. Light. Ice, 1*3060 1-3073 Red Reusch. Iceland-spar 1*65844 1*48639 D 24** v. d. WiUigen. Nitrate of Soda, 1*5854 1*3369 D 23** F. Kohlransoh. Quartz, 1-64419 1*55329 D 24** v. d. Willigen. Tourmaline, 1*6479 1*6262 Green 22° Heusser. Zircon, 1*92 1*97 Red de Senarmont. Biaocal Crystals, THBEB PRINCIPAL INDICES OF BEFRACTION FOB SODIUM LIGHT. Least. Intermediate. Greatest. Temp. Observer. Arragonite, 1*53013 1*68157 1*68589 Rudberg. Borax, 1*4463 1*4682 1*4712 23° Kohlrausch. Mica, 1*5609 1*5941 1*5997 23° Nitre, 1*3346 15056 15064 16° Schrauf. Selenite, 1*52082 1*52287 1*53048 17° v. Lang. ^5?fs^tic)}l-9S«5 2*0383 2*2405 16° Schrauf. Topaz, 1*61161 1*61375 1*62109 Rudberg. INDICES OF BEFRACTION FOR LIQUIDS. 93. The following values of indices of refraction for liquids are condensed from Fraunhofer's determinations, as given by Sir John Herschel (" Enc. Met. Art.," Light, p. 415) :— fTafer, density 1*000. B 1*3309; C 1*3317; D 1'3336; E 1 3358 ; F 1 3378 ; G 1-3413 ; H 1*3442. F 82 UNITS AND PHYSICAL CONSTANTS. [chap. OU of Turpmtine, density 0*8a5. B 1-4705; C 1-4715; D 1-4744; E 1-4784; F 1-4817; G 1 •488-2; H 1-4939. The following determinations of the refractive indices of liquids are from Gladstone and Dale's results, as given in Watf 8 " Dictionary of Chemistry," iii pp. 629-631 :— Sulphide of Carbon^ at temperature 11', A 1-6142; B 1-6207; C 16240; D 1-6333; E 1-6465; F 1-6584; G 1*6836; H 1-7090. Benzene, at temperature 10*5^ A 1-4879; B 1-4913; C 1-4931; D 1-4976; E 1-5036; F 15089; G 15202; H 1-5305. Chlorofomij at temperature 10**. A 1-4438; B 14457; C 1-4466; D 1-4490; E 1-4526; F 1-4555; G 14614; H 1-4661. Alcohol^ at temperature 15°. A 1-3600; B 13612; C 13621; D 13638; E 1-3661; F 1.3683; G 1-3720; H 1-3751. EiMr, at temperature 15°. A 1-3529; B 13545; C 1-3554; D 1-3666; E 1-3590; F 1-3606; G 13646; H 1-3683. Watery at temperature 15**. A 1-3284; B 1-3300; C 13307; D 13324; E 1-3347; F 1-3366; G 1*3402; H 1*3431. INDICES FOR OASES. 94. Indices of refraction of air at 0** C. and 760 mm. for the principal fraunhofer lines. vra.] LIGHT. 83 Acoordliig to Kettler. According to Lorenz. A 1*00029286 1*00028935 B 29350 28993 C 29383 29024 D 29470 29108 E 29584 29217 F 29685 29312 G 29873 29486 H 30026 29631 95. The formula established by the experiments of Biot and Arago for the index of i*efraction of air at various pressures and temperatures was _ . ^ -0002943 Ji ^ Y+at * 760' a denoting the coefficient of expansion '00366, and h the pressure in millims. of mercury at zero. As the pressure of 760 millims. of such mercury at Paris is 1*0136 x 10^ dynes per sq. chl, the general formula applicable to all localities alike will be ,_ -0002943 P ^ l + *00366< • 10136 xlO«' where P denotes the pressure in dynes per sq. cm. This can be reduced to the form •0002903 P /*-! 1 + oosae^ ' W 96. According to Mascart, /a - 1 for any gas is pro- portional not to = but to h + ph^- where fi and a' are coefficients which vary from one gas to another. In the following table, the column headed fl^, 84 UNITS AND PHYSICAL CONSTANTS. [chap. contains the indices for 0"* and 760 mm. at Paris. The next column contains the value of P multiplied bj 10^ (it being understood that h is expressed in millimetres), and the next column the value of a'. All these data are for the light of a sodium flame :— Ih /5xW a' Air, 1'0002927 7*2 -00382 Nitrogen, 2977 8*5 382 Oxygen, 2706 IM Hydrogen, 1387 -8-6 378 Nitrous Oxide, 5159 88 388 Nitrous Gas, 2975 7 367 Carbonic Oxide, 3350 8*9 367 Carbonic Acid, 4544 72 406 Sulphurous Acid, .... 7036 25 460 Cyanogen, 8216 277 More recent, and probably more accurate observations, which will be published in vol. v. of " Travaux et Memoires du Bureau International des Poids et Mesures/' have been conducted by Benoit with Fizeau's dilatometer. They give 1-0002923 as the index of refraction of air for the D line at 0" 0. and 760 mm.; and for the temperature coefficient they give •003667, which is identical with the coefficient of expansion of air. The larger value, •00382, obtained by Mascart, is traced to imperfect measurement of temperature. Coefficient of Dispersive Power, 97. Assuming Cauchy's formula /.-l=a(l + ^) vm.] LIGHT. 85 (where A is the wave-length), which is known to be approximately true for air within the limits of the visible spectrum, the constant b may be called the coefficient of dispersive power. Employing as the unit of length for A the 10""* of a centimetre, Mascart ("Ann, de V l^le Normale/' 1877, p. 62) has obtained the following values for b: — Coefficient of Dispersion. Air, -0058 Nitrogen, -0067 Oxygen, "0064 Hydrogen, 0043 Carbonic Oxide, '0075 Carbonic Acid, 0052 Nitrous Oxide, 0125 Cyanogen, 0100 According to Mascart, the ratio of dispersion to devia- tion for the two lines B and H is *024 for air, *032 for the ordinary ray in quartz, '038 for light crown glass, *040 for water, and *046 for the ordinary ray in Iceland- spar. Motation of Plane of Poh/rization, 98. The rotation produced by 1 millim. of thickness of quartz cut perpendicular to the axis has the following values for different portions of the spectrum, according to the observations of Soret and Sarasin (" Com. Ben. 95," p. 635, 1882), the temperature of the quartz being 20* C. :— Rotation. A 12'-668 B 15^-746 C 17**-318 D, 2r-684 Di 2r-727 Rotation. E 27"-543 F 32**-773 G 42"-604 H 6ri93 86 UNITS AND PHYSICAL CONSTANTS. [cHAP.vni. According to the same observers, the rotation at f C. is equal to the rotation at 0*" C. multiplied by 1+-000179<. Units of Illuminating Power. 99. The British '^ Candle" is a spermaceti candle, { inch in diameter (6 to the lb.), burning 120 grains per hour. The French " Carcel " is a lamp of specified construc- tion, burning 42 grammes of pure Colza oil per hour. One " carcel " is equal to about 9 J " candles." The unit adopted by the International Congress at Paris, April 1884, is a square centimetre of molten platinum at the temperature of solidification. The surface illuminated by it in photometric tests is to be normally opposite to the surface of the molten platinum. Accord- ing to the experiments of M. Yiolle the author of this unit, it is equal to 2*08 carcels. It is therefore about 19| candles. 87 CHAPTER IX. HEAT. 100. The unit of lieat is usually defined as the quantity of heat required to raise, by one degree, the temperature of unit mass of water, initially at a certain standard temr perature. The standard temperature usually employed is 0** 0. ; but this is liable to the objection that ice may be present in water at this temperature. Hence 4** C. has been proposed as the standard temperature ; and another proposition is to employ as the unit of heat one hundredth part of the heat required to raise the unit mass of water from 0** to 100' C. 101. According to Kegnault (" M6m. Acad. Sciences," xxi. p. 729) the quantity of heat required to raise a given mass of water from 0** to t"" C. is proportional to «+ -000 02^2 + -000 000 3^3. . . . (i) The mecm thermal capacity of a body between two stated temperatwres is the quantity of heat required to raise it from the lower of these temperatures to the higher, divided by the difference of the temperatures. The mean thermal capacity of a given mass of water between 0° C. and f is therefore proportional to 1 + -000 02< + -000 000 3^2. ... (2) 88 UNITS AND PHYSICAL CONSTANTS. [chap. The thermal capacity of a body at a started temperaJtwre is the limiting value of the mean thermal capacity as the range is indefinitely diminished. Hence the thermal capacity of a given mass of water at f is proportional to the differential coefficient of (1), that is to 1 + OOO 04^ + -000 000 9<2. ... (3) Hence the thermal capacities at 0^ and 4* are as 1 to 1*000174 nearly; and the thermal capacity at 0* is to the mean thermal capacity between 0** and 100"* as 1 to 1-005. 102. If we agree to adopt the capacity of unit mass oi water at a stated temperature as the unit of capacity, the unit of heat must be defined as n times the quantity of heat required to raise unit mass of water from this initial temperature through — of a degree when n is indefinitely n great. Supposing the standard temperature and the length of the degree of temperature to be fixed, the units both of heat and of thermal capacity vary directly as the unit of mass. In what follows, we adopt as the unit of heat (except where the contrary is stated) the heat required to raise a gramme of pure water through 1*" C. at a temperature intermediate between 0** and 4**. This specification is sufficiently precise for the statement of any thermal measurements hitherto made. 103. The thermal capaciti/ of unit mass of a substance at any temperature is called the specific heat of the sub- stance at that temperature IX.] HEAT. 89 Specific heat is of zero dimensions in length, mass, and i^ime. It is in fact the ratio increment of heat in the substance increment of heat in water ibr a given increment of temperature, the comparison being between eqital masses of the substance at the actual temperature and of water at the standard tempei^ture. "The numerical value of a given concrete specific heat merely depends upon the standard temperature at which the specific heat of water is called unity. 104. The thermal capacity of unit volume of a sub- stance is another important element : we shall denote it by e. Let s denote the specific heat, and d the density of the substance ; then c is the thermal capacity of d units •of mass, and therefore c=^sd. The dimensions of c in lengthy mass, and time are the same as those of c?, namely, M Y^. Its numei'ical value will not be altered by any change in the units of length, mass, and time, which leaves the value of the density of water unchanged. In the O.G.S. system, since the density of water between 0° and 4" is very approximately unity, the thermal capacity of unit volume of a substance is the value of the ratio increment of heat in the substance increment of heat in water for a given increment of temperature, when the compari- json is between equal volumes, 105. Mr. Herbert Tomlinson ("Proc. Roy. Soc," June 19, 1885) has obtained the following deteiminations of specific heat from observations conducted in a uniform 90 UNITS AND PHYSICAL CONSTANTS. [chap. manner with metallic wires well annealed. The wires were heated sometimes to 60*" C. and sometimes to 100*" C, and were plunged in water at 20**. The formulse are for the true specific heat at f C : — Aluminium, "20700 + •0002304< Copper, -09008 + -0000648^ German Silver, -09413 + -0000106^ Iron, -10601 + -000 140« Lead, 02998 + -00003U Platmum, -03198 + 'OOOOIS^ Platinum Silver, -04726 + •000028^ Silver, -05466 + -000044< Tin, -05231 + -000072* Zinc, -09009 + -000075< The formulse for the mean specific heat between 0* and t"* are obtained from these by leaving the first term un- changed and halving the second term. YioUe has made the following determinations of specific heat at f : — Platinum, -0317 + -000012* Iridium, -0317 + -000012* Palladium, -0582+ -000020* H. F. Weber has determined the specific heat of diamond to be •0947 + -000 994* - -000 000 30*2, and consequently the mean specific heat of diamond from 0** to *• to be •0947 + -000 497* - -000 000 12*2. The mean specific heat of ice according to Kegnault is •504 between - 20* and 0^ and -474 between - 78** and 0'. 106. The following list of specific heats of elementary substances is condensed from that given in Landolt and Bomstein's tables : — IX.] HEAT. Substance. Temperature. Aluminium, 16'to 97" Antimony, 13 ,, 106 Arsenic (crystalline), 21,, 68 „ (amorphous), 21 ,, 65 Bismuth, 9 „ 102 Borax (crystalline), „ 100 ,, (amorphous), 18,, 48 Bromine, solid, -78 ,,-20 ,, liquid, 13 „ 45 Cadmium, 0,, 100 Calcium, 0„ 100 Carbon, diamond, 11 » graphite, 11 ,, wood charcoal,.. to 99 Cobalt, 9 „ 97 Copper, 15 „ 100 Gold, „ 100 Iodine, 9 „ 98 Iridium, „ 100 Iron, 60 Lead, 19 to 48 Lithium, 27 „ 99 Magnesium, 20 ,, 51 Manganese, 14 ,, 97 Mercury, solid, -78 ,,-40 „ liquid, 17 „ 48 Molybdenum, 6 ,, 15 Nickel, 14 „ 97 Osmium, 19 „ 98 Palladium, „ 100 Phosphorus (yellow,8olid) -78 ,, 10 ( „ liquid) 49 „ 98 (red), 15 „ 98 Platinum, „ 100 Potassium, -78 „ 91 Sp. Heat. Observer. •2122 Regnault. •0486 B^de. •0830 J Bettendorflf & } WiUlner. •0758 »» »» •0298 B6de. •2518 Mixter&Dana •254 Kopp. •0843 Regnault. •1071 Andrews. •0548 Bunsen. •1804 it •1128 H. F. Weber. •1604 •1935 >» •1067 Regnault. •0933 B6de. •0316 VioUe. •0541 Regnault. •0323 VioUe. •1124 Bystrom. •0315 Kopp. •9408 Regnault. •245 Kopp. •1217 Regnault. •0319 >> •0335 Kopp. •0659 De la Rive and Marcet. • •1092 Regnault. •0311 »» •0592 Violle. •1699 Regnault. •2045 Person. •1698 Reg^ult. •0323 Violle. •1655 Regnault. 92 UNITS AND PHYSICAL CONSTANTS. [chap. Sttbttanoe. Temperature. 8p. Heat. Otieerver. Rhodium, 10 „ 97 *0580 BQgnaalt Selenium, crystalline, 22 „ 62 -0840 j ^w^f * SiHcon, crystalline, 22 '1697 H. F, Weber. Silver, OtolOO "0559 Bunsen. Sodium, -28 „ 6 "2934 Begnauli Sulphur (rhomb, cryst), 17 „ 45 *163 Kopp. „ (newly melted), 15 „ 97 '1844 Begnault. Tellurium, crystalline,.... 21 „ 51 *0475 Kopp. Thallium, 17 „ 100 "0335 Regnault. Tin, cast, „ 100 -0559 Bunsen. Zinc, „ 100 -0935 „ Svhatcmcea not Elementary, Brass (4 copper 1 tin), hard, 15''to98^ *0858 Begnault „ „ soft, 14 „ 98 '0862 „ Ice, -20 „ -504 „ 107. The following determinations of specific heat oJ liquids are by Kegnault. We have omitted decixna figures after the fourth, as even the second figure ii •different with different observers : — Alcohol. Chloroform. Oil of Turpentine. Temp. Sp. Ht. Temp. 8p. Ht. Temp. Sp. Ht -20** -6063 -30* -2293 -20** 3842 '5475 -2324 '4106 40 '6479 30 '2354 40 '4538 80 '7694 60 '2384 80 '4842 120 '5019 160 '5068 Ether. Bisulphide of Carbon. Temp. Sp. Ht Temp. Sp. Ht. -30' '5113 -30'* '2303 '5290 2352 30 '5468 30 '2401 Schiiller has found the specific heat of liquid benzine at ^ to be •37980+ -001 44^. IX.] HEAT. 9a 108. The following table (from Miller's ''Chemical Phyaics/' p. 308) contains the results of Kegnault's ex- periments on the specific heat of gases. The column headed " equal weights '' contains the specific Jisats in the sense in which we have defined that term. The column headed " equal volumes " gives the relative thermal capa- cities of equal volumes at the same pressure and tem- perature : — Thermal Capacities of Gases and Vapours at Cotistant Fress'ure, Gas or Vapour, Air, - Oxygen, - - Nitrogen,- Hydrogen, Chlorine, - Bromine, - Nitrous Oxide, Nitric Oxide, - Carbonic Oxide Carbonic \ Anhydride, ) Carbonic Di-\ sulphide, / Ammonia, Marsh Gas, defiant Gas, - Arsenious ^ Chloride, / Silicic Chloride Titanic Stannic Sulphurous \ -^oihydride,/ 9> Equal Vola. •2375 •2405 •2368 •2359 •2964 3040 3447 •2406 2370 3307 4122 •2996 3277 4106 7013 •7778 8564 8639 341 Weights. •2375 •2176 •2438 3-4090 •1210 •0555 •2262 •2317 •2450 •2163 •1569 •5084 •5929 •4040 •1122 •1322 •1290 •0939 •154 Gas or Vapour. Hydrochloric \ Acid, - -j" Sulphuretted \ Hydrogen, / Steam, Alcohol, - Wood Spirit, - Ether, - - Ethyl Chloride, Bromide, Disul- \ dphide, j yanide, CMoroform, - Dutch Liquid, Acetic Ether, - Benzol, Acetone, - Oil of Turpen- tine, - Phosphorus Chloride Equal Vols. »> } ,} •2352 •2857 •2989 •7171 •5063 1 -2266 •6096 •7026 1 -2466 •8293 •6461 •7911 1-2184 1-0114 -8341 2-3776 •6386 Weij(ht8 1842 2432 4805 4534 4580 I 4796 2738 1896 4008 4261 1566 2293 4008 3754 4125 5061 1347 94 UNITS AND PHYSICAL CONSTANTS. [chap. 109. E. Wiedemann (** Pogg. Ann.," 1876, No. 1, p. 39) has made the following determinations of the specific heats of gases : — Specific HeaL AtO'. Air, 0-2389 Hydrogen, 3*410 Carbonic Oxi<le, 0-2426 Carbonic Acid 0*1952 Ethyl, 0-3364 Nitric Oxide, 0-1983 Ammonia, 0*6009 Multiplying the specific heat by the relative density, he obtains the following values of Thermal Capacity of Equal Volumes. At 100*. At 200*. Relative Density. 1 0-0692 • • * • ■ • • • • • • • • • • 0-967 0-2169 0-2387 1*629 0*4189 0-5015 0*9677 0-2212 0-2442 1-5241 0*5317 0-5629 0-5894 At 100*. At 200'. At 0\ Air, 0*2389 Hydrogen, 0*2359 Carbonic Oxide, . . 0*2346 Carbonic Acid,... 0*2985 Kthyl, 0*3254 Nitric Oxide, 0*3014 Ammonia, 0*2952 The same author ("Pogg. Ann.," 1877, New Series, vol. ii. p. 195) has made the following determinations of specific heats of vapours at temperature f : — Specific Heat. 0*3316 0-3650 0-4052 0-4851 0-3362 0-3712 0-3134 0-3318 VaTxiui- ^^S'^ ^^ Temp. \ ap*)ui . jjj Experiments. Chlorofonu, -JO'-g to 189'*-8 Bromic Ethyl,.. 27°-9 to 189^-5 Benzine, 34°-l to 115*^1 Acetone, 26" -2 to 179° '3 Acetic Ether, ... 32*'-9 to 113''-4 Ether, 26° -4 to 188° '8 •1341 -f 0001354/ 1354 + 003560^ •2237 + -0010228/ •2984 -f -0007738/ -2738 + -0008700^ •.S725 + 0008536/ IX.] HEAT. 95 Kegnault's determinations for the same vapours were as follows : — Vapour. Bange of ^ Temperature. Mean Specific Heat for this Bange. Chloroform, 117° to 228" Bromic Ethyl, ... 77" 7 to 1 96" '5 Benzme, 116" to 218" Acetone, 129" to 233" Acetic Ether,. ... 115" to 219" Ether, 70" to 220" According to Regnault. •1567 •1896 •3754 •4125 •4008 •4797 According to Wiedemann. •1573 •1841 •3946 •3946 •4190 •4943 Regnault has also determined the mean specific heat of bisulphide of carbon vapour between 80** and 147* to be •1534, and between 80' and 229* to be -1613. Melting Points and Heat op Liquefaction. 110. VioUe has made the following determinations of melting points (" Com. Ren.," Ixxxix. p. 702) : — Silver, 954'* Gold, 1045 Copper, 1054 Palladium, 1500' Platinum, 1775 Iridium, 1950 This last temperature 1950* is very near to that of the hottest part of the oxyhydrogen flame. The same observer has found the latent heat of lique- fJEiction of platinum to be 27*2, and of palladium 36*3 ("Com. Ren," Ixxxv. p. 543, and Ixxxvii. p. 981). 111. The following approximate table of melting points is based on that given in the second supplement to Watt's " Dictionary of Chemistry," pp. 242, 243 :— Platinum, 2000° Palladium, 1950 Ck)ld, 1200 Castlron, 1200 Glass, 1100 Copper, 1090° SUver, 1000 Borax, 1000 Antimony, 432 Zinc, 360 96 UNITS AND PHYSICAL CONSTANTS. [chap. Lead, 330* Cadmimn, 320 Bismuth, 265 Tin, 230 Selenium 217 Cane Sugar, 100 Sulphur, Ill Sodium,... 90 Wax, 68** Potassium, 58 Paraffin, 54 Spermaceti, 44 Phosphorus, 43 Water,* Bromine, -21 Mercury, -40 Melting Latent Point. Heat. Silver, 1000* 211 Zinc, 433 281 Chloride of Calcium (CaC1.3H20),.... 28-6 40-7 Nitrate of Potas- sium, 339 47-4 Nitrate of Sodium, 310*5 63*0 112. The following table (from Watt's "Dictionary of Chemistry," vol. iii. p. 77) exhibits the latent heats of fluidity of certain substances, together with their melting points : — Melting Latent Point. Heat. Mercury, -39" 2-82 Phosphorus, . . 44 5*0 Lead, 332 5*4 Sulphur, 115 9-4 lodme, 107 117 Bismuth, 270 12-6 Cadmium, 320 13-6 Tin, 235 14-25 The latent heat of fluidity of water was found by Regnault, and by Provostaye and Desains, to be 79*. Bunsen, by means of his ice-calorimeter (** Pogg. Ann.," vol. cxli. p. 30), has obtained the value 80-025. He finds the specific gravity of ice to be '9167. 113. Chandler Roberts and Wrightson have compared the densities of molten and solid metals by weighing a solid metal ball in a bath of molten metal either of tLe same or a different kind (" Phys. Soc," 1881, p. 195, and 1882, p. 102). They find that "iron expands rapidly (as much as 6 per cent.) in cooling from the liquid to the plastic state, and then contracts 7 per cent, to solidity ; whereas bismuth appears to expand in cooling from the liquid to the solid state about 2-35 per cent." The following is their tabular statement of results : — IX.] HEA*r. 97- MetaL Sp. Grav. ofSoUd. Bismuth, 9*82 Copper, 8*8 Lead, 11-4 Tin, 7-5 Zinc, 7-2 SUver, 10-57 Iron (No. 4 foundry, Cleveland), 6-95 8p. Grav. of Liquid. 10055 8-217 10-37 7-025 6-48 9-51 6-88 Percentage of change in volume trtntk. cold solid to liquid. Decrease of vol. 2*3 Increase of vol. 7-1 9-93 6-76 11-1 11-2 1-02 114. Change of volume in melting, from Kopp^s experi- ments (Watt's " Die," Art. Heat, p. 78) :— Phosphorus, Calling the volume at O*" unity, the volume at the melting point (44**) is 1*017 in the solid, and 1-052 in the liquid, state. Svlphur. Volume at 0° being 1, volume at the melting point (115**) is 1-096 in the solid, and 1*150 m the liquid, state. Wax. Volume at 0* being 1, volume at melting point (64*) is 1'161 in solid, and 1-166 in liquid, state. Stearic Acid, Volume at 0" being 1, volume at melting point (70°) is 1079 in soUd, and 1-198 in Uquid, state. Rost*8 Fusible Afetal (2 parts bismuth, 1 tin, 1 lead). Volume at 0** being 1, volume at 59° is a maximum, and is 1-0027. Volume at melting point (between 95° and 98°) is greater in liquid than in solid state by 1 '55 per cent. 115. The following table (from Millers "Chemical Physics," p. 344) exhibits the change of volume of certain substances in passing from the liquid to the vaporous condition under the pressure of one atmosphere : — 1 volume of water yields 1696 volumes of vapour, alcohol 528 ether 298 oil of turpentine 193 O )> »» »» n >» »> »» »» } ' 98 UNITS AND PHYSICAL CONSTANTS. [chap. 116. The following table of boiling points and heats of vajjorization, at atmospheric pressure, is condensed from Landolt and Bornstein, pp. 189, 190 : — BoUing Latent Heat of ob««rver Point. Vaporisation. "«Morver. Alcohol, 77*9 202-4 Andrews. Bisulphide of Carbon, 46*2 86*7 Bromine, 58 46*6 ,, Ether, 34*9 90-4 „ Mercury, 350 62-0 Person. Oil of Turpentme, 159-3 74-0 Brix. Sulphur, 316 362-0 Person. Water, 100 535-9 Andrews. 117. Kegnaiilt's approximate formula for what he calls 'Hhe total heat of steam at ^°/' that is, for the heat required to raise unit mass of water from C to ^ in the liquid state and then convert it into steam at Cy is 606-5 + 'Z^ht. If the specific heat of water were the same at all tempera- tures, this would give 606-5 -•695« as the heat of evaporation at f. But since, according to Regnault, the heat required to raise the water from 0° to € is t + 000 02^' + -000 000 3«3, the heat of evaporation will be the difference between this and the " total heat," that is, will be 606-5 - •695« - -000 02«2 _ -000 DOO 3<3, which is accordingly the value adopted by Kegnault as the heat of evaporation of water at f, 118. According to Regnault, the increase of pressure at constant volume, and increase of volume at constant IX.] HEAT. 99 pressure, when the temperature increases from 0° to 100°, haTe the following values for the gases named : — Q At Constant At Constant Volume. Pressure. Hydrogen -3667 '3661 Air, -3665 3670 Nitrogen, 3668 Carbonic Oxide, 3667 3669 Carbonic Acid, 3688 '3710 Nitrous Oxide, 3676 3719 Sulphurous Acid, 3845 '3903 Cyanogen, 3829 3877 Jolly has obtained the following values for the coeffi- cient of increase of pressure at constant volume : — Air, 00366957 Oxygen, 00367430 Hydrogen, 00365620 Nitrogen, 0036677 Carbonic Acid, 0037060 Nitrous Oxide, 0037067 Mendelejeff and Kaiander have determined the co- efficient of expansion of air at constant pressure to be •0036843. 119. Kegnault's results as to the departures from Boyle's law are given in the form — -^l|i=l±A(»»-l)±B(m-in Vj denoting the volume at the pi^essure P^, Vq the volume at atmospheric pressure Pq, and m the ratio ^. For air, the negative sign is prefixed to A and the posi- tive sign to B, and we have log A = 3-04351 20, log B = 5-2873751. 100 UNITS AND PHYSICAL CONSTANTS. [chap. For nitrogen, the signs are the same as for air, and we have log A = 4-8399376, log B = 6-8476020. For carbonic acid, the negative sign is to be prefixed both to A and B, and we have log A = 3-9310399, log B = 6-8624721. For hydrogen, the positive sign is to be prefixed both to A and B, and we have log A = 4-7381 736, log B = 6-9250787. 120. The following table, showing the maximum pres- sure of aqueous vapour at temperatures near the ordinary boiling point, is based on Eegnault's determinations, as revised by Moritz (Guyot's Tables, second edition, collec- tion D, table xxv.) : — Dynes per sq. cm. 9-779 xlO» 9-814 Temperature. Centims. of Mercury at Paris. 99 0" 73-319 99-1 73-584 99-2 73-849 99-3 74-115 99-4 74-382 99-5 74-650 99-6 74-918 99-7 75-187 99-8 75-457 99-9 75-728 100-0 76-000 1001 76-273 100-2 76-546 100-3 76-820 »» »f » J 9-849 9-885 9-920 9-956 9 992 1 0028 X W 1-0064 „ 1-0100 ,» 1-0136 „ 1-0173 „ 1-0209 ,, 1-0245 „ a.] HEAT. 101 Temperature. Centims. of Mercury at Paris. Dynes per sq. cm. 100-4 77-095 1-0282x106 100 6 77-371 1-0319 „ 100-6 77-647 1-0356 „ 100-7 77-925 1-0393 „ 100-8 78-203 1-0430 „ 100-9 78-482 1-0467 „ 101-0 78-762 1-0605 „ 121. Mdximum Pressure of Aqueous Vapowr at various temperatures, in dynes per sq, centim. -20** -15 1236 1866 -10 2790 - 6 4150 6133 5 10 15 8710 12220 16930 20 25 30 40 23190 31400 42050 73200 50' l-226xl0«^ 60 1-985 „ 80 4-729 „ 100 1014x106 120 1-988 „ 140 3-626 „ 160 6-210 „ 180 1006x107 200 1-560 Ifaadmum Pressure of various Vapours, in dynes per sq. cm. AlcoboL Ether. Sulphide of Carbon. Chloroform. -20** 4455 9-19 X W 6-31 X 10* -10 8630 1-53 X W 1 -058 X 10* 16940 2-46 „ 1-706 „ 10 32310 3-826 „ 2-648 „ 20 59310 5-772 „ 3-975 „ 2141 X W 30 1-048 X 10» 8-468 „ 5-799 „ 3-301 „ 40 1-783 „ 1-210 X 10« 8-240 „ 4927 „ 50 2-932 „ 1-687 „ 1-144 X 10« 7-14 „ 60 4-671 „ 2-301 „ 1-554 „ 1-007 X 10« 80 1 -084 X 106 4 031 „ 2-711 „ 1-878 „ 100 2-265 „ 6-608 „ 4-435 „ 3-24 „ 120 4-31 „ 1 029 X 107 6-87 „ 5-24 „ 102 UNIT8 AND PHYSICAL CONSTANTS. [chap. 122. The following are approximate values of the maximum pressure of aqueous vapour at various tempera- tures, in millimetres of mercury. They can be reduced to dynes per sq. cm. by multiplying by 133*4 : — O*' mm. 4-6 92^* mm. 567 112° mm. 1150 132° mm. 2155 10 9-2 94 611 114 1228 134 2286 20 17-4 96 658 116 1311 136 2423 30 31-5 98 707 118 1399 138 2567 40 54-9 100 760 120 1491 140 2718 50 96-2 102 816 122 1588 142 2875 60 149 104 875 124 1691 144 3040 70 233 106 938 126 1798 146 3213 80 355 lOS 1004 128 1911 148 3393 90 525 110 1075 130 2030 150 3581 123. The density (in gm. per cub. cm.) of aqueous vapour at any temperature t and any pressure jt? (dynesper sq. cm.), whether equal to or less than the maximum pressure, is •622 X -001276 p X ^ 1 + •00366« 10«* If q denote the pressure in millimetres of mercury, the approximate formula is •622 X -001293 o 1 X •00366< 760 124. Temperature of evaporation and dew-point (Glaisher's Tables, second edition, page iv.). The fol- lowing are the factors by which it is necessary to mul- tiply the excess of the reading of the dry thermometer over that of the wet, to give the excess of the tempera- ture of the air above that of the dew-point : — Beading of Beading of Dry Bulb Therm. Factor. Dry Bulb Pacto Therm. -10°C.=14°F. 8-76 15°C. = 59°F. 1-89 - 5 23 7-28 20 68 1-79 32 3-32 25 77 1-70 + 5 41 2-26 30 86 1-65 + 10 50 2-06 35 95 1-60 IX.] HEAT. 103 125. Critical temperatures of gases, above which thej cannot be liquefied (abridged from Landolt and Bomstein, p. 62) :— Critical Max. Pressure Temperature. of Gas at this Observer, o Temp. Hydrogen, - 174*2 98 '9 atm. Sarrau . Oxygen, -105-4 487 „ Nitrogen, -123*8 421 „ Carbonic Acid, 30*92 ,, Andrews. „ 32-0 77-0 „ Sarrau. Bisulphide of Carbon, 271*8 74*7 ,, Sajotschewsky. Sulphurous Acid, 155*4 78*9 ,, ,, Chloroform,.... 260*0 64*9 ,, ., Benzol, 280*6 49*5 ,, ,, Alcohol, 234*3 62*1 „ Ether, 1900 36*9 „ Conductivity. 126. By the tliermal conductivity of a substance at a given temperature is meant the value of k in the expression q^kA^^:i^H, (1) X where Q denotes the quantity of heat that flows, in time t, through a plate of the substance of thickness Xy the area of each of the two opposite faces of the plate being A, and the temperatures of these faces being respectively i?, and V2» ®^ch differing but little from the given temper- ature. The lines of flow of heat are supposed to be normal to the faces, or, in other words, the isothermal Bui*faces within the plate are supposed to be parallel to the faces ; and the flow of heat is supposed to be steady, in other words, no part of the plate is to be gaining or losing heat on the whole. 104 UNITS AND PHYSICAL CONSTANTS. [chap. Briefly, and subject to these understandings, conduc- tivity may be defined as the quantity of heat that passes in unit tiiney through unit area of a plate whose thickness is u/nityy when its opposite faces differ in temperature by one degree, 127. Dimensions of Conductivity, From equation (1) we have ]q— Q ^ /2) v^ - Ti * A< The dimensions of the factor — ^ — are simply M, since the unit of heat varies jointly as the unit of mass and the length of the degree. The dimensions of the factor - — are =r- ; hence the dimensions of k are --— . This is At LiT ij^ on the supposition that the unit of heat is the heat required to raise unit mxiss of water one degree. In calculations relating to conductivity it is perhaps more usual to adopt as the unit of heat the heat required to raise unit volume of water one degree. The dimensions of Q . . . . L2 will then be L^, and the dimensions of h will be 7p • i?2 - 1? ' T These conclusions may be otherwise expressed by say- M ing that the dimensions of conductivity are =r-= when the liT thermal capacity of unit mass of water is taken as unity, and are — when the capacity of unit volume of water is taken as unity. In the C.G.S. system the capacities of unit mass and unit volume of water are practically identical. IX.] HEAT. 105 n 128. Let c denote the thermal capacity of unit volume of a substance through which heat is being conducted. Then - denotes a quantity whose value it is often neces- sary to discuss in investigations relating to the transmis- sion of heat. We have, from equation (2), c v^-Vj^' At^ Ok where Q' denotes — . Hence - would be the numerical c c value of the conductivity of the substance, if the unit of heat employed were the heat required to raise unit volume of the substance one degree. Professor Clerk Maxwell k proposed to call - the thermometric conductivity, as dis- c tinguished from k the thermal or calorimetric conductivity. "We prefer, in accordance with Sir Wm. Thomson's article, k " Heat," in the Encychpcedia Britannica^ to call - the diffusimty of the substance for heat, a name which is k based on the analogy of - to a coefficient of diffusion. Coefficient of Diffusion, 129. There is a close analogy between conduction and diffusion. Let x denote the distance between two parallel plane sections A and B to which the diffusion is perpendicular, and let these sections be maintained in constant states. Then, if we suppose one substance to be at rest, and another substance to be diffusing through it, the coefficient of diffusion K is defined by the equation y=4 (1) 106 UNITS AND PHYSICAL CONSTANTS. [chap. where y denotes the thickness of a stratum of the mixture as it exists at B, which would be reduced to the state existing at A by the addition to it of the quantity which diffuses from A to B in the time t When the thing diffused is heat, the states at A and B are the temperatures v^ and v^ and y denotes the thickness of a stratum at the lower temperature which would be raised to the higher by the addition of as much heat as passes in the time t. This quantity of heat, for unit area, will be which must therefore be equal to yc{v^ - v^\ whence we have k t y = - -. c X k The " thermometric conductivity " - may therefore be re- c garded as the coefficient of diffusion of heat. 130. When we are dealing with the mutual inter- diffusion of two liquids, or of two gases contained in a closed vessel, subject in both cases to the law that the volume of a mixture of the two substances is the sum of the volumes of its components at the same pressure, the quantity of one of the substances which passes any section in one direction must be equal (in volume) to the quantity of the other which passes it in the opposite direction, since the total volume on either side of the section remains unaltered ; and a similar equality must hold for the quantities which pass across the interval between two sections, provided that the absorption in the interval IX.] HKAT. 107 itself is negligible. Let x as before denote the distance between two parallel plane sections A and B to which the* diffasion is perpendicular. Let the mixture at A consist of m parts by volume of the first substance to 1 - m of the second, and the mixture at B consist of n parts of the second to 1 - n of the first, m being greater than 1 - w, and therefore n greater than 1 -m. The first substance will then diffuse from A to B, and the second in equal quantity from B to A. Let each of these quantities be such as would form a stratum of thickness z (the vessel being supposed prismatic or cylindrical, and the sections considered being normal sections), then z will be propor- tional to • 7n-(l-n) , , . ^ m + n-l ^ -t, that IS to L X X and the coefficient of interdiffusion K is defined by the equation ==K'?i+'*-ri< (2> X The numerical quantity m + 7i - 1 may be regarded as measuring the difference of states of the two sections A and B. If y now denote 4)he thickness of a stratum in the con-^ dition of B which would be reduced to the state existing at A by the abstraction of a thickness z of the second substance, and the addition of the same thickness of the first, we have {\-n)y-\-z as the expression for the quantity of the first substance in the stratum after the operation. This is to be equal to my. Hence we have 108 UNITS AND PHYSICAL CONSTANTS. [chap. And sabstituting for z its value in (2) we have finally y = K-, X (4) which is of the same form as equation (1), ^ now denoting the thickness of a stratum of the mixture as it exists at B, which would be reduced to the state existing at A by the addition to it of the quantity of one substance which diffuses from A to B in the time t^ and the removal from it of the quantity of the other substance which diffuses from B to A in the same time. 131. The following values of K in terms of the centi- metre and second are given in Professor Clerk Maxwell's ** Theory of Heat," 4th edition, p. 332, on the authority of Professor Loschmidt of Vienna. Coefficients of Interdiffu^cfn of Gases Carbonic Acid and Air, Hydrogen, Oxygen,.... 1423 5614 1409 1586 1406 0982 7214 1802 6422 4800 k These may be compared with the value of - for air, which, according to Professor J. Stefan of Vienna, is '256. The value of k for air, according to the same authority, is 6*58 X 10"", and is independent of the pressure. Pro- fessor Maxwell, by a different method, calculates its value At 5-4 X lO-''. Marsh Gas, Carbonic Oxide, Nitrous Oxide,.. Oxygen and Hydrogen, ,, ,, Carbonic Oxide, Carbonic Oxide and Hydrogen, Sulphurous Acide and Hydrogen,... IX.] HEAT. 109 Results of Experim&rUs on Condticiivity of Solids, 132. Principal Forbes' results for the conductivity of iron (Stewart on Heat, p. 261, second edition) are ex- pressed in terms of the foot and minute, the cubic foot of water being the unit of thermal capacity. Hence the value of Forbes* unit of conductivity, when referred to C.G.S., is (30-48)' 60 or 15*48; and his results must be multiplied by 15*48 to reduce them to the C.G.S. scale. His observations were made on two square bars ; the side of the one being 1^ inch, and of the other an inch. The results when reduced to C.G.S. units are as follows : — Temp. Cent. 25 50 75 100 125 150 175 200 225 250 276 l^-inch bar. 1-incb bar. -207 1536 -1912 1460 *1771 -1399 *1656 -1339 -1567 1293 *1496 1259 -1446 *1231 -1399 -1206 -1366 *1183 -1317 *1160 -1279 -1140 -1240 -1121 133. Neumann's results ("Ann. de. Chim. 185) must be multiplied by '000848 to reduce them to our scale. They then become as follows : — Copper, 1*108 Brass, -302 Zinc, -307 Iron, •164 German Silver, '109 Ice '0057 " vol. Ixvi. p. 110 UNITS AND PHYSICAL CONSTANTS. [chap. In the same paper he gives for bhe following substances k k , the values of -3 or - ; that is, the quotient of conductivity by the thermal capacity of unit volume. These require the same reducing factor as the values of k^ and when reduced to our scale are as follows : — Values of -. Coal, •ooiie'' Melted Sulphur, -00142 Ice, -0114 Snow, -00356 Frozen Mould, 00916 Sandy Loam, -0136 Granite (coarse), -0109 Serpentine, -00594 134. Sir W. Thomson's results, deduced from observa- tions of underground thermometers at three stations at Edinburgh ("Trans. R. S. E.," 1860, p. 426), are given in terms of the foot and second, the thermal capacity of a cubic foot of water being unity, and must be multiplied by (30-48)2 or 929 to reduce them to our scale. The following are the reduced results : — it, or /• Conductivity. ^- Trap-rock of Calton Hill, -00415 -00786 Sand of experimental garden, -00262 '00872 Sandstone of Craigleith Quarry, 01068 -0231 1 My own result for the value of -- from the Greenwich underground thermometers ("Greenwich Observations," 1860) is in terms of the French foot and the year. As a French foot is 32-5 centims., and a year is 31557000 seconds the reducing factor is (32-5)2-^ 31557000; that is, 3-347 X 10-^ The result is a- c* Gravel of Greenwich Observatory Hill, -01249 IX.] HEAT. Ill Professors Ayrton and Perry ("Phil. Mag.," April, 1878) determined the conductivity of a Japanese building stone (porphyritic trachyte) to be '006 9. 1 35. Angstrom, in " Pogg. Ann.," vols. cxiv. (1861) and cxviii (1863), employs as units the centimetre and the minute ; hence his results must be divided by 60. These results, as given at p. 294 of his second paper, will then stand as follows : — Value of -. c Copper, first specimen, 1-216 (1 - -00214 t) „ second specimen, 1 '163 (1 - -001519 t) Iron, -224 (1 - '002874 t) He adopts for c the values -84476 for copper ; -88620 for iron, and thus deduces the following values of k : — Conductivity. Copper, first specunen, 1 027 (1 - '00214 t) „ second specimen, -983 (1 - -001519 t) Iron •199(1-002874 136. A Committee, consisting of Professors Herschel and Lebour, and Mr. J. F. Dunn, appointed by the British Association to determine the thermal conductivities of certain rocks, have obtained results from which the following selection has been made by Professor Herschel : — Substance. ^?P^"«^HT*^? ^ - C.G.S. Units. c Iron pyrites, more than -01 more than '01 70 Rock salt, rough crystal, '0113 -0288 Fluorspar, rough crystal, 00963 '0156 Quartz, opaque crystal, and quartzites, '0080 to -0092 '0175 to 0190 Siliciou88andstone8(8lightlywet), '00641 to -00854 '0130 to -0230 112 UNITS AKD PHYSICAL CONSTANTS. [chap. 8»bd.nc,. ^l^uJl!'' I Galena, rough crystal, inter- spersed with quartz, •00705 •0171 Sandstone and hard grit, dry, ... -00545 to '00565 '0120 Sandstone and hard grit, thor- oughly wet, -00590 to -00610 -0100 Micaceous flagstone, along the cleavage, -00632 "OllO Micaceous flagstone, across cleav- age, -00441 -0087 Slate, along cleavage, 00550 to -00650 0102 Do., across cleavage, -00315 to '00360 -0057 Granite, various specimen8,aboat -00510 to -00550 -OlOOto -012D Marbles, limestone, calcite, and compact dolomite -00470 to -00560 -0085 to -0095 Red Serpentine (Cornwall), -00441 -0065 Caen stone (building limestone), -00433 -0089 Whinstone, trap rock, and mica schist, -00280 to '00480 -0055 to -0095 Clay slate (Devonshire), -00272 -0053 Tough clay (sun-dried), -00223 -0048 Do., soft (with one-fourth of its weight of water), 00310 -0035 Chalk, 00200 to -00330 0046 to -0059 Calcareous sandstone (firestone), '00211 -0049 Plate-glass German and English, '00198 to -00234 -00395 to ? German glass toughened, '001 85 -00395 Heavy spar, opaque rough crystal, '00177 i'ire-brick, -00174 0053 Fine red brick, '00147 -0044 Fine plaster of Paris, dry plate, -00120 '^^^^\ab t Do., thoroughly wet, 00160 -0025/ White sand, dry, -00093 -0032 Do. , saturated with water, about 00700 -0120 about House coal and cannel coal, '00057 to -00113 *0012 to *0027 Pumice stone, -00055 IX.] HEAT. 113 137. P6clet in " Annales de Chimie," s6r. 4, torn. ii. p. 114 [1841], employs as the unit of conductivity the tmns- mission, in one second, through a plate a metre square and a millimetre thick, of as much heat as will raise a cubic decimetre (strictly a kilogramme) of water one degree. Formula (2) shows that the value of this conduc- tivity in the C.G.S. system, is '''' ±.; that is, ' 1 10000' '100 His results must accordingly be divided by 100. The same author published in 1853 a greatly extended series of observations, in a work entitled "Nouveaux documents relatifs aux chauffage et k la ventilation." In this series the conductivity which is adopted as unity is the transmission, in one hour, through a plate a metre square and a metre thick, of as much heat as will raise a kilogramme of water one degree. This conductivity, in C.G.S. units is 1000 100 1 ., ^ . 1 ; that IS, - 1 10000 3600' '360 The results must therefore be divided by 360. Those of them which refer to metals appear to be much too small. The following are for badly conducting substances : — Density. Conductivity. Fir, across fibres, -48 -00026 „ alongfibres, -48 '00047 Walnut, across fibres, "00029 ,, alongfibres, -00048 Oak, across fibres, '00059 Cork, -22 -00029 Caoutchouc, -00041 Guttapercha, -00048 Starch paste, 1017 '00118 Glass, 2-44 -0021 H 114 UNITS AND PHYSICAL CONSTANTS. [chap. Density. Glass, 2-66 Sand, quartz, 1*47 Brick, pounded, coarse-grained, . 1 '0 „ passed through ( , .-« silk sieve,.... ( Fine brickdust, obtained by decan- \ I -- tation, j ^ °^ Chalk, powdered, slightly damp, '92 washed and dried, '85 washed, dried, and \ , ^o compressed, j^ "^ Potato-starch, '71 Wood-ashes, '45 Mahogany sawdust, '31 Woodcharcoal,ordinary,powdered, '49 Bakers' breeze, in powder, passed \ .qk through silk sieve, / >* i* •41 Conductivity. •0024 00075 00039 00046 00039 00030 00024 00029 00027 00018 00018 00022 00019 Ordinary wood charcoal in powder, j .^ passea through silk sieve, \ Coke, powdered, '77 Iron filings, 2*05 Biuoxide of manganese, 1 '46 Woolly Substances, Cotton Wool, of all densities, Cotton swansdown (molleton de \ coton), of all densities j Calico, new, of all densities, Wool, carded, of all densities, ... Woollen swansdown (molleton de \ laine) of all densities, J Eider-down, Hempen cloth, new, '54 „ old, '58 Writing-paper, white, '85 Grey paper, unsized,... '48 138. In Professor George Forbes's paper on conductivity ^" Proc. R. S. E.," February, 1873) the units are the centim. and the minute; hence his results must be divided by 60. 000225 00044 00044 00045 000111 000111 000139 000122 000067 000108 000144 000119 000119 000094 IX.] HEAT. 115 In a letter dated March 4, 1884, to the author of this work, Professor Forbes remarks that the mean tempera- ture of the substances in these experiments was - 10°, and expresses the opinion that bad conductors (such as most of these substances) conduct worse at low than at high temperatures — an opinion which was suggested by the analogy of electrical insulators. His results reduced to C.G.S. are— 00223 00213 00177 00115 00081 00072 000717 0005 000453 000405 000335 0003 *000088 Kamptulikon, .... Vulcanized india- rubber, Horn, Beeswax, Felt,.... Vulcanite, Haircloth, Cotton- wool, divided, „ pressed, Flannel, Coarse linen, Quartz, along axis, } »» »> a Ice, along axis, Ice, perpendicular to\ axiB, / Black marble, White marble, Slat«, Snow, Cork, Glass, Pasteboard, Carbon, Eoofing-felt, Fir,- parallel to fibre. Fir, across fibre, and\ . along radius, / Boiler-cement, -000162 Paraffin, -00014 Sand, very fine, -000131 Sawdust, -000123 Professor Forbes quotes a paper by M. Lucien De la Rive ("Soc. de Ph. et d'Hist. Nat de Geneve," 1864) in which the following result is obtained for ice, Ice, -00230. M. De la Rive's experiments are described in " Annales de Chimie," ser. 4, tom. i. pp. 504-6. 139. Dr. Robert Weber (** Bulletin, Soc. Sciences Nat. de Neufchatel," 188), has found the following conductivi- ties and surface emissivities for five specimens of rock from the St. Gothard tunnel : — Quartz,perpendicular \ to axis, J 00011 000 089 000087 000 087 -000087 000 0833 0000402 000 0433 000 0335 0000355 000 0298 000922 00124 00057 00083 0040 0044 116 UNITS AND PHYSICAL CONSTANTS. [chap. Specimen No. IQS.—MiccLceous Gneiss, Conductivity, -000917 + 0000044^ Emissivity, -000185 + '0000023^ Specific Heat, -1778 + -00042^ Specimen No. 114. — Mica Schist, Conductivity, '000733 + '000 010^ Emissivity, -000207 + 'OOOOOIB^ Specific Heat, -18000 + 00044^ Specimen No. 124. — Eurite. Conductivity, '000862 + •00016^ Emissivity, -000249 + '00000009^ Specific Heat, '1682 + 'OOOee Specimen No. 140. — Gneiss, Conductivity, -0014 + -000003^ Emissivity, -00026 + •0000008< - Specific Heat, -1463 + •0009< Specimen No. 146. — Micaceous Schist, Conductivity, '000952 + '000 009^ Emissivity, '000168 + '0000023^ Specific Heat, '1697 + '0006^ Conductivity of Liquids, 140. The conductivity of water, according to experi- ments by Mr. J. T. Bottomley ("Phil. Trans." 1881, April 3), is '002, which is nearly the same as the con- ductivity of ice. (See 138.) 141. H. F. Weber ("Sitz. kon. Preuss. Akad." 1885), has made the following determinations of conductivities of liquids at temperatures of from 9** to 15*" C. He em- ploys the centimetre, the gi-amme, and the minute as units: we have accordingly divided the original numbers by 60 to reduce to C.G.S. rx.] HEAT. 117 Conduc- tivity. Water, -00136 Aniline, '000408 Glycerine, '000670 Ether, '000303 Methyl Alcohol, , Ethyl Alcohol,... Propyl Alcohol, . Butyl Alcohol,.., Amyl Alcohol,... Ameisen Acid,..., Acetic Acid, Propion Acid,.... Butyric Acid, ...., Isobutyric Acid, . Valerian Acid,... Isovalerian Acid,, Isocapron Acid, . . , Methyl Acetate,. . Ethyl Formiate,.., Ethyl Acetate, Propyl Formiate, . Propyl Acetate, .., Methyl Butyrate,, Ethyl Butyrate, . . Methyl Valerate,. Ethyl Valerate,... 000495 000423 000373 000340 000328 000648 000472 000390 000360 000340 000325 000312 000298 000386 000378 000348 000357 000327 000335 000318 000315 000307 Amyl Acetate, Chloro Benzol, Chloroform, Chloro Carbon, Propyl Chloride, .. Isobutyl Chloride,. Amyl Chloride, Bromo Benzol, Ethyl Bromide, Propyl Bromide, Isobutyl Bromide,... Amyl Bromide, Ethyl Iodide,. Propyl Iodide, Isobutyl Iodide,. Amyl Iodide,.... Benzol, Toluol, Cymol, Oil of Turpentine, .... Sulphuric Acid, Bisulphide of Carbon, Oil of Mustard, Ethyl Sulphide, Conduc- tivity. •000302 000302 000288 000252 000283 000278 000284 000265 000247 000257 000278 000237 000222 000220 000208 000203 000333 000307 000272 000260 000765 000343 000382 000328 In the original paper these numbere are compared with the thermal capacities of the liquids per unit of volume, and with the calculated mean distances between their molecules. It is found that conductivity, multiplied by mean distance, divided by capacity, is a nearly constant quantity for the members of any one of the above groups. Comparing one group with another, its most widely dif- ferent values are represented by 19 and 23, if we except the last group, for which its value is between 26 and 27. Amission and Surface Conduction. 142. Mr. D. M'Farlane has published (" Proc. Roy. Soc." 118 UNITS AND PHYSICAL CONSTANTS. [chap. 1872, p. 93) the results of experiments on the loss of heat from blackened and polished copper in air at atmospheric pressure. They need no reduction, the units employed being the centimetre, gramme, and second. The general result is expressed by the formulae X = -000238 + 3-06 X IQ-H - 2-6 x lO'^^^ for a blackened surface, and i»= -000168 + 1 -98 X 10-«^- 1-7 x IQ-^t:' for polished copper, x denoting the quantity of heat lost per second per square centim. of surface of the copper, per degree of diflference between its temperature and that of the walls of the enclosure. These latter were blackened internally, and were kept at a nearly constant temperature of 14° C. The air within the enclosure was kept moist by a saucer of water. The greatest difference of tempera- ture employed in the experiments (in other words, the highest value of t) was 50° or 60° C. The following table contains the values of x calculated from the above formulae, for every fifth degree, within the limits of the experiments : — Diflference of Value of X. Ratio. Temperarure. shed Surface. Blackened Surface. 5° •00017S •000252 •707 ! 10 •000186 •000266 •699 ' 15 •000193 •000279 •692 , 20 •000-201 •000289 •695 ' 25 •000207 •000298 •694 30 •000212 •000306 •693 35 •000217 •090313 •693 1 40 000220 •000319 •693 1 45 000223 •000323 •690 ; 50 000225 •000326 •690 55 000226 •000328 •690 60 000226 •000328 •690 IX.] HEAT. 119 143. Professor Tait has published (" Proc. R. S. E." 1869-70, p. 207) observations bv Mr. J. P. Nichol on the loss of heat from blackened and polished copper, in air, at three different pressures, the enclosure being blackened internally and surrounded by water at a temperature of approximately 8° C* Professor Tait's units are the grain- degree for heat, the square inch for area, and the hour for time. The rate of loss per unit of area is heat emitted area x time ' The grain-degree is "0648 gramme-degree. The square inch is 6*4516 square centims. The hour is 3600 seconds. Hence Professor Tait's unit rate of emission is •0648 6-4516 X 3600 = 2-79xlO-« of our units. Employing this reducing factor, Professor Tait's Table of Results will stand as follows : — Pressure 1 '014 x 10* [760 niillims. of mercury]. Blackeued. Temp. Cent. Less per sq. cm. ^ per second. 61-2 -01746 50-2 -01360 41-6 -01078 34-4 -00860 27-3 -00640 20-5 -00455 Bright. Temp. Cent. Loss per sq. cm. o per second. 63-8 -00987 57-1 -00862 50-5 00736 44-8 -00628 40-5 -00562 34-2 -00438 29-6 -00378 23-3 -00278 18-6 -00210 •This temperature is not stated in the " Proceedings" but has been communicated to me by Professor Tait. 120 UNITS AND PHYSICAL CONSTANTS. [chap. Pressure 1*36 x 10* [102 millims. of mercury]. 62-5 01298 57-0 -01168 53-2 -01048 47-5 -00898 43 -00791 28-5 -00490 67-8 -00492 61-1 -00433 65 -00383 49-7 -00340 44-9 -00302 40-8 00268 Pressure 1*33 x 10* [10 millims. of mercury]. 62-6 -01182 57-5 -01074 54-2 -01003 65 -00388 60 -00355 50 -00286 41-7 -00726 40 -00219 37-5 -00639 30 -00157 34 00569 23-5 -00124 27-5 00446 24-2 -00391 I Mechanical Equivalent of Heat. 144. The value originally deduced by Joule from his experiments on the stirring of water waa 772 foot-pounds of work (at Manchester) for as much heat as raises a pound of water through 1° Fahr. This is 1389-6 foot- pounds for a pound of water raised 1° C, or 1389-6 foot- grammes for a gramme of water raised 1° C. As a foot is 30-48 centims., and the value of g at IManchester is 981-3, this is 13896 x 30-48 x 981-3 ergs per gramme- degree; that is, 4-156 x 10^ ergs per gramme- degree. A later determination by Joule (" Brit. Assoc. Report," 1867, pt. i. p. 522, or "Reprint of Reports on Electrical Standards," p. 186) is 25187 foot-grain-second units of work per grain-degree Fahr. This is 45337 of the same units per grain-degree Centigrade, or 45337 foot-gramme- second units of work per gramme-degree Centigrade; that is to say, 45337 X (30-48)2 = 4-212x10' ergs per gramme-degree Centigrade. IX.] HEAT. 121 In view of the fact that the B. A. standard of electrical resistance employed in this determination is now known to be too small by about 1 '3 per cent., and that the cur- rent energy converted into heat was accordingly under- estimated to this extent, the result ought now to be in- creased by 1*3 per cent, which will make it 4-267 X 107. At the meeting of the Royal Society, January, 1878 (" Proceedings," vol. xxvii. p. 38), an account was given by Joule of experiments recently made by him with a view to increase the accuracy of the results given in his former paper. ("Phil. Trans.," 1850.) His latest result from the thermal effects of the friction of water, as announced at this meeting, is, that taking the unit of heat as that which can raise a pound of water, weighed in vacuo, from 60* to 61° of the mercurial Fahrenheit thermometer; its mechanical equivalent, reduced to the sea-level at the latitude of Greenwich, is 772*55 foot-pounds. To reduce this to water at 0* C. we have to multiply by 1-00089,* giving 773*24 ft. lbs., and to reduce to ergs per gramme-degree Centigrade we have to multiply by 981-17 X 30-48 x?. 5 The product is 4-1624 x 10^. 145. Some of the best determinations by various experi- menters are given (in gravitation measure) in the following list, extracted from "Watts' Dictionary of Chemistry," Supplement 1872, p. 687. The value 429*3 in this list corresponds to 4-214 x 10^ ergs : — * This factor is found by giving t the value 15*8 (since the tem- perature 60*5 Fahr. is 15 '8 Cent.) in formula (3) of art 101. 122 UNITS AND PHYSICAL CONSTANTS. [chap. Hini| i» )) »» Joule,... Violle,... 432 Friction of water and brass. 433 Escape of water under pressure. 441*6 Specific heats of air. 425''2 Crushing of lead. ^Q.o 5 Heat produced by an electric I current. 435-2 (copper).. . \ 434*9 (aluminium) I Heat produced by induced cur- 435-8 (tin) ( rents. 437*4 (lead) ) Regnault, 437 Velocity of sound. We shall adopt 4*2 x 10*^ ergs as the equivalent of 1 gramme-degree ; that is, employing J as usual to denote Joule^s equivalent, we have J = 4*2x 107 = 42 millions. 146. Heat and Energy of Combination with Oxygen, 1 gramme of Compound formed. Gramme- degrees of heat produced. Equivalent Energy, in erga. Hvdrofiren C02 S02 P206 ZnO FeW Sn02 CuO C02 CO^andH^O >» )) 34000 AF 8000 A F 2300 A F 5747 A 1301 A 1576 A 1233 A 602 A 2420 A 13100 A F 11900 A F 6900 A F 1-43x1012 3-36x1011 9-66 X 101® 2-41 X 10" 5-46x1010 6-62x1010 518 „ 2-53 „ 1-02x10" 5-50 „ 5-00 „ 2-90 „ Carbon, Sulphur, Phosphorus, Zinc, Iron, Tin, -^ **^» • Copper, Carbonic oxide,.... Marsh-sras defiant gas, Alcohol, Combustion in Chlorine. Hydrogen, . Potassium, Zinc, Iron, Tin, Copper, .... HCl KCl ZnCP Fe2Cl« SnCl^ CuCP 23000 F T 9-66x10" 2655 A 1-12 „ 1529 A 6-42x1010 1745 A 7-33 „ 1079 A 4-53 „ 961 A 404 „ IX.] HEAT. 123 The numbers in the last column are the products of the numbers in the preceding column hy 42 millions. The authorities for these determinations are indicated by the initial letters A (Andrews), F (Favre and Silber- mann), T (Thomsen). Where two initial letters are given, the number adopted is intermediate between those obtained by the two experimenters. 147. Difierence between the two specific heats of a gas. Let 8^ denote the specific heat of a given gas at con- stant pressure, ^2 the specific heat at constant volume, a the coefficient of expansion per degree. V the volume of 1 gramme of the gas in cubic centim. at pressure ^ dynes per square centim. "When a gramme of the gas is raised from 0** to 1° at the constant pressure ^^, the heat taken in is s^, the increase of volume is av, and the work done against external resist- ance is avp (ergs). This work is the equivalent of the difierence between s^ and Sg ; that is, we have «j - ^2 = ^» w^eie J = 4-2 X 207. For dry air at 0** the value of vp is 7*838 x 10^, and a is '003665. Hence we find 5j - Sg = '0684. The value of «p according to Regnault, is '2375. Hence the value of «2 is •1691. The value of ^~ ^ , or -f*, for dry air at 0° and a megadyne per square centim. is ?i.7 *2 = :2^ = 8-728 X 20-^ : V 783-8 ] 24 UNITS AND PHYSICAL CONSTANTS. [chap. ^nd this is also the value of J — ? for any other gas (at the same temperature and pressure) which has the same •coefficient of expansion. 148. CJhange of freezing point due to change of pressure. Let the volume of the substance in the liquid state be to its volume in the solid state of 1 to 1 + 6. When unit volume in the liquid state solidifies under pressure P+p, the work done by the substance is the product of P +p by the increase of volume e, and is there- fore Te+pe, If it afterwards liquefies under pressure P, the work Hlone against the resistance of the substance is Pe ; and if the pressure be now increased to P +p, the substance will be in the same state as at first. Let T be the freezing temperature at pressure P, T + t the freezing temperature at pressure T +p, I the latent heat of liquefaction, d the density of the liquid. Then d is the mass of the substance, and Id is the heat taken in at the temperature of melting T. Hence, by thermodynamic principles, the heat converted into mecbani- •cal effect in the cycle of operations is TT273 • ^' But the mechanical effect is pe. Hence we have T+273 J I _ e(T+273) . ,„> ~p Jld~ <^^ nc.] HEAT. 125 — - is the lowering of the freezing-point for an additional pressure of a dyne per square centim.; and — x 10^ will be the lowering of the freezing point for each addi- tional atmosphere of 10^ dynes per square centim. For water we have e=-087, Z= 79-25, T = 0, rf=l, - 1 X 1 0« = ?f ^ l^ll = -007 14. p 42 X 79-25 Formula (3) shows that — is opposite in sign to e. P Hence the freezing point will be raised by pressure if the substance contracts in solidifying. 149. Change of temperature produced by adiabatic com- pression of a fluid ; that is, by compression under such circumstances that no heat enters or leaves the fluid. Let a cubic centim. of fluid at the initial temperature ^* C. and pressure p dynes per square centim. be put through the cycle of operations represented by the annexed "indicator diagram," ABCD, where horizontal distance from left to right denotes increase of volume and perpen- dicular distance upwards increase of pressure. In AD let the pressure be constant and equal to p. In BC let the pressure be constant and equal to ;? + tt, tt being small. Let AB and CD be adiabatics, so near together that AD and BC are very small compared with the altitude of the figure which is TT. 126 UNITS AND PHYSICAL CONSTANTS. [cjhap. The figure will be ultimately a parallelogram, so that the changes of volume AD and BC will be equal ; let their common value be called edt, e denoting the expansion per degree at constant pressure; dt will therefore be the difference of temperature between A and D, or between B and C. We suppose this difference to be very small compared with the difference of temperature between A and B or between C and D. The cycle is reversible ; let it be performed in the direc- tion ABCD. Then heat is taken in as the substance expands from B to C, and given out as it contracts from D to A. The work done by the substance in the cycle is equal to the area of the parallelogram, which, being the product of the base edt by the height tt, is Tredt, The heat given out in DA is Cdt, C denoting the thermal capacity of a cubic centim. of the substance at constant pressure; hence the " efficiency " is — , and this, by the rules of J O T Thermodynamics, must be equal to — ^ , where t de- notes the increase of temperature from A to B. Put T for the absolute temperature 273 + <, then we have T^Te where t is the increase of temperature produced by the increase tt of pressure. 150. Resilience as affected by heat of compression. The expansion due to the increase of temperature t, TTre^ above calculated, is re ; that is, -=— ; and the ratio of J O IX.] HEAT. 127 this expansion to the contraction — , which would be pro- duced at constant temperature (E denoting the resilience ETe^ of volume at constant temperature), is -^p, : 1- Putting m for , the resilience for adiabatic compression will be J \j E ; or, if m is small, E (1 +m) ; and this value is to 1 - w be used instead of E in calculating the change of volume due to sudden compression. The same formula expresses the value of Young's modulus of resilience, for sudden extension or compression of a solid in one direction, E now denoting the value of the modulus at constant temperature. Eocafth'pleB, For compression of water between 10** and 11° we have E = 2-1 X low, T = 283, e = -000 092, C = 1 ; hence ^^=-0012. For longitudinal extension of iron at 10** we have E= 1-96 X 10^^ T = 283, e= -000 0122, C = -109 x 77 ; hence ^^ = -00234. Thus the heat of compression increases the volume- resilience of water at this temperature by about \ per cent., and the longitudinal resilience of iron by about \ per cent. 128 UNITS AND PHYSICAL CONSTANTS. [chap. For dry air at 0° and a megadyne per square centim., we have E = 10«, T = 273, e - }--, C = -2375 x -001276, m=__ = .288. \—m = 1 404. 151. Eocpansions of Volumes per degree Cent, {abridged from WaM " Dictionary of Chemistry,** Article Heat, pp. 67, 68, 71). Glass -00002 to -00003 Iron -000035 „ '000044 Copper, -000052 ,, -000057 Platmum, -000026 ,, -000029 Lead, -000084 „ -000089 Tin,....., -000058 „ -000069 Zinc, -000087 „ 000090 Gold -000044 „ -000047 Brass, -000053,, -000056 Silver, -000057 „ -000064 Steel, -000032 „ -000042 Cast Iron, about '000033 These results are partly from direct observation, and partly calculated from observed linear expansion. Expansion of Mercury, according to Regnault ( Watts* ^* Dictionary,'* p. 56). Temp. = t. o . 10 . 20 . 30 . 50 . 70 . 100 . Volume at t. Expansion per degi*ee at t". 1-000000 -00017905 1-001792 -00017950 1-003590 -00018001 1-005393 '00018051 1-009013 -00018152 1-012655 '00018253 1-018153 '00018405 The temperatures are by air-thermometer. IX.] HEAT. 129 The formula adopted by the Bureau International des Poida et Meaures for the volume at f C. (derived from Regnault's results) is 1 + •000181792«+ -000 000 000 175^2 + •000 000 000 035116^^. Expansion of Alcohol and Ether, according to Kojjp {Watts' '* Dictionary ," p. 62). Volume. Temp. Alcohol. Ether. 6 1-0000 10000 10 1-0105 10152 20 10213 10312 30 1-0324 10483 40 1-0440 1-0667 152. Collected Data for Dry Air, Expansion from 0° to 100° at const, pressure, as 1 to 1 '367 or as 273 to 373 Specific heat at constant pressure, '2375 „ ,, at constant volume, -1691 Pressure-height at 0° C, about 7*99 x 10* cm., or about 26210 ft. Standard barometric column, 76 cm. = 29 '922 inches. Standard pressure, 1033*3 gm. per sq. cm. or 14-7 lbs. per sq. inch. or 2117 lbs. „ foot. or 1 '0136 X 10^ dynes per sq. cm. Standard density, at 0" C, -001293 gm. per cub. cm. or -0807 lbs. per cub. foot. Standard bulkiness, 773 -3 cub . cm. per gm. or 12-39 cub. ft. per lb. I 1 30 UNITS AND PHYSICAL CONSTANTS, [chap. ix. Di*y and Moist Air, Mass of 1 Cubic Metre in Qrammes. Temp.C. Dry Air. Saturated Air. slto^tion. 6 1293-1 1290-2 4-9 10 1247*3 12417 9*4 20 1204-6 1194-3 17*1 30 1164-8 1146-8 30*0 40 1127-6 1097-2 507 If A denote the density of dry air and W that of vapour at 3 saturation, the density of saturated air is A - - W, or more exactly A - 608 W. 131 CHAPTER X. MAGNETISM. 153. The unit magnetic pole, or the pole of unit strength, is that which repels an equal pole at unit distance with unit force. In the C.G.S. system it is the pole which repels an equal pole, at the distance of 1 centimetre, with a force of 1 dyne. If P denote the strength of a pole, it will repel an equal p2 pole at the distance L with the force — -. Hence we have the dimensional equations P2L-2 = force = MLT-2, p2 = ML^T'S, P = M^L^T"'; that is, the dimensions of a pole (or the dimensions of strength of pole) are M*L^'~\ 154. The work required to move a pole P from one point to another is the product of P by the difference of the magnetic potentials of the two points. Hence the dimensions of magnetic potential are ^^- = ML^-- . M ~ 4l ~ 'T = m4l4t-\ 155. The vrUensity of a magnetic ^'eW is the force which a unit pole will experience when placed in it. Denoting 132 UNITS AND PHYSICAL CONSTANTS. [chap. this intensity by I, the force on a pole P will be IP. Hence IP = force = MLT-^ I = MLT-2. M'4l-^T = M*L"4t'^ ; that is, the dimensions o^ field-intensity are M*L" ^T"^ 156. The moment of a magnet is the product of the strength of either of its poles by the distance between them. Its dimensions are therefore LP; that is, M*LTr"\ Or, more rigorously, the moment of a magnet is a quantity which, when multiplied by the intensity of a uniform field, gives the couple which the magnet ex- periences when held with its axis perpendicular to the lines of force in this field. It is therefore the quotient of a couple ML^T-^by a field-intensity M^L"*!"^; that is, it is M^L^T"^ as before. 157. If different portions be cut from a uniformly mag- netized substance, their moments will be simply as their volumes. Hence the intensity of m,ag7ietization of a uni- formly magnetized body is defined as the quotient of its moment by its volume. But we have moment ^ m^L^t-i . L"^ = M^L " ^T'^ volume Hence intensity of magnetization has the same dimensions as intensity oi field. When a magnetic substance (whether paramagnetic or diamagnetic) is placed in a magnetic field, it is magnetized by induction, and the ratio of the intensity of the magnetization . thus produced to the intensity of the field is called the " coeflicient of magnetic X.] MAGNETISM. 133 induction," or " coeflficient of induced magnetization," or the "magnetic susceptibility" of the substance. For paramagnetic substances (such as iron, nickel, and cobalt) this coefficient is positive; for diamagnetic substances (such as bismuth), it is negative ; that is to say, the induced polarity is reversed, end for end, as compared with that of a paramagnetic substance placed in the same field. 158. It has generally been stated that " magnetic sus- ceptibilty " is nearly independent of the intensity of the field so long as this intensity is much less than is required for saturation. But R. Shida found ("Proc. Roy. Soc," Nov., 1882), in the softest iron wire, a very rapid varia- tion of susceptibility at low intensities. Under the influence of the earth's vertical force at Glasgow, '545, the susceptibility had the very large value 734 when the wire was stretched by a weight, and 335 when the weight was off. Under a magnetizing force 2*35, the susceptibilities, with and without the weight, were 235 and 154. Saturation was obtained with a magnetizing force of 80*7, which produced magnetizations 1390 and 1430, the susceptibilities being therefore 17*1 and 17*6. With pianoforte wire (steel), the susceptibilities were 67*5 and 69*3 under the earth's vertical force, and 13*2 when saturation was just attained, with a magnetizing force of 107*5. The magnetization at saturation was 1420, being about the same as for soft iron wire. With a square bar of soft iron nearly 1 centim. square, the susceptibility diminished from 19, under a magnetizing force of 18*2, to 7*6, under a magnetizing force of 189, which just produced saturation. 134 UNITS AND PHYSICAL CONSTANTS. [chap. \ Examples, 1. To find the multiplier for reducing magnetic in- tensities from the foot-grain-second system to the C.G.S. system. The dimensions of the unit of intensity are M^L~^T'\ In the present case we have M = -0648, L = 30-48, T = 1, since a grain is *0648 gramme, and a foot is 30*48 centini. Hence M*L"iT~' = ^^11 = -04611; that is, the foot- grain-second unit of intensity is denoted by the number •0461 1 in the C.G.S. system. This number is accordingly the required multiplier. 2. To find the multiplier for reducing intensities from the millimetre-milligramme-second system to the C.G.S. system, we have 1000 10 Hence - is the required multiplier. 3. Gauss states (Taylor's " Scientific Memoirs," vol. ii. p. 225) that the magnetic moment of a steel bar-magnet, of one pound weight, was found by him to be 100877000 millimetre-milligramme-second units. Find its moment in C.G.S. units. Here the value of the unit moment employed is, in terms of C.G.S. units, M^L^'T-^ where M is 10"^ L is 10-^ and T is 1 ; that is, its value islO~*.10~^^ = 10"**. Hence the moment of the bar is 10087*7 C.G.S. units. X.] MAGNETISM. 135 4. Find the mean intensity of magnetization of the bar, assuming its specific gravity to be 7*85, and assuming that the pound mentioned in the question is the pound avoir- dupois of 453*6 grammes. Its mass in grammes, divided by its density, will be its volume in cubic centimetres ; hence we have 453-6 7-65 = 57*78 = volume of bar. X . .. n i.- i.- moment 10088 ^^a £* Intensity of magnetization = — =-=r=-=T» = 174*0. •^ ^ volume 57*78 5. Kohlmusch states (" Physical Measurements," p. 195, English edition) that the maximum of permanent mag- netism, which very thin rods can retain is about 1000 millimetre-milligramme-second units of moment for each milligramme of steel. Find the corresponding moment per gramme in C.G.S. units, and the corresponding in- tensity of magnetization. For the moment of a milligramme we have 1000 X 10-4 =10-1. For the volume of a milligramme we have (7*85)-^ X 10-^, taking 7 '85 as the density of steel. Hence the moment per gramme is 10"^ x 10^= 100, and the intensity of magnetization is 100 x 7*85 = 785. 6. The m^aximtim intensity of magnetization for speci- mens of iron, steel, nickel, and cobalt has been deter- mined by Professor Rowland ("Phil. Mag.," 1873, vol. xlvi. p. 157, and November, 1874) — that is to say, the limit to which their intensities of magnetization would approach, if they were employed as the cores of electro- magnets, and the strength of current and number of con- volutions of the coil were indefinitely increased. Professor 136 UNITS AND PHYSICAL CONSTANTS. [chap. Rowland's fundamental units are the metre, gramme, and second ; hence his unit of intensity is — ^ of the C.G.S. unit. His values, reduced to C.G.S. units, are At 12''C. At 220'C. Iron and Steel, 1390 1360 Nickel, 494 380 Cobalt, 800 (?) 7. Gauss states (loc. cit) that the magnetic moment of the earth, in millimetre-milligramme-second measure, is 3-3092 R3, R denoting the earth's radius in millimetres. Reduce this value to C.G.S. Since R^ is of the dimensions of volume, the other factor, 3*3092, must be of the dimensions of intensity. Hence, employing the reducing factor 10"^ above found, we have '33092 as the corresponding factor for C.G.S. measure ; and the moment of the earth will be •33092 R3, R denoting the earth's radius in centimetres — that is 6-37 X 108. We have •33092 X (6-37 x 108)3 := 8-55 x 1025 for the eartKs magnetic moinent in C.G.S. units. 8. From the above result, deduce the intensity of mag- netization of the earth regarded as a uniformly magnetized body. We have . . .. moment 8*55 x lO^^ ^-^^ ''^''"'•*y =vd^ = 1-083 710^ ="^^^°- This is about of the intensity of magnetization of 2200 -^ ° X.] MAGNETISM. 137 Gauss's pound magnet; so that 2 '2 cubic decimetres of earth would be equivalent to 1 cubic centim. of strongly magnetized steel, if the observed effects of terrestrial mag- netism were due to uniform magnetization of the earth's substance. 9. Gauss, in his papers on terrestrial magnetism, em- ploys two different units of intensity, and makes mention of a third as ** the unit in common use." The relation between them is pointed out in the passage above referred to. The total intensity at Gottingen, for the 19th of July, 1834, was 4*7414 when expressed in terms of one of these units — the millimetre-milligramme-second unit ; was 1357 when expressed in terms of the other unit em- ployed by Gauss, and 1*357 in terms of the "unit in common use." In C.G.S. measure it would be *47414. 159. A first approximation to the distribution of mag- netic force over the earths s surface is obtained by assuming the earth to be uniformly magnetized, or, what is mathe- matically equivalent to this, by assuming the observed effects to be due to a small magnet at the earth's centre. The moment of the earth on the former supposition, or the moment of the small magnet on the latter, must be *33092 R^ R denoting the earth's radius in centims. The magnetic poles, on these suppositions, must be placed at 77" 50' north lat., 296" 29' east long., and at 77" 60' south lat., 116" 29' east long. The intensity of the horizontal component of terrestrial magnetism, at a place distant A" from either of these poles, will be 33092 sin A" : 138 UNITS AND PHYSICAL CONSTANTS. [chap. the intensity of the vertical component will be '66184 cos A"; and the tangent of the dip will be 2 cotan A°. The magnetic potential, on the same supposition^ will be •33092 ?L cos A', r being variable. (See Maxwell, " Electricity and Mag- netism," vol. ii. p. 8.) Gauss's approximate expression for the potential and intensity at an arbitrary point on the earth's suiface consists of four successive approximations, of which this is the first. 160. According to " Airy on Magnetism," the place of greatest horizontal intensity is in lat. 0° long. 259** E., where the value is '3733 ; the place of greatest total in- tensity is in South Victoria, about 70° S., 160° E., where its value is '7898, and the place of least total intensity is near St. Helena, in lat. 16° S., long. 355° E., where its value is -2828. 161. The following mean values of the magnetic ele- ments at Greenwich have been kindly furnished by the Astronomer Royal (Dec, 1885) : — West Declination, IS** 15' 0- (<- 1883) x T'SO, Horizontal Force, 0-1809 + (<- 1883) x -00018. Dip, 67° 3r-8-(^- 1883) X 1-39. Vertical Force, 0-4374 -U- 1883) x -00007. = Horizontal force x tan. dip. Each of these formulae gives the mean of the entire year t, 162. According to J. E. H. Gordon ("Phil. Trans.," 1877, with correction in "Proc. Roy. Soc," 1883, pp. X.] MAGNETISM. 139 4, 5), the rotation of the plane of polarisation between two points, one centimetre apart, whose magnetic potentials (in C.G.S. measure) differ by unity, is (in circular mea- sure) 1-52381x10-^ in bisulphide of carbon, for the principal green thallium ray, and is 2-248 X 10-6 in distilled water, for white light. Mr. Gordon infers from BecquereFs experiments ("Comp. Rend.," March 31, 1879) that it is about 3 X 10-^ for coal gas. According to Lord Rayleigh (" Proc. Roy. Soc./' Dec. 29, 1884), the rotation for sodium light in bisulphide of carbon at 18* C. is -04202 minute. This is 1-22231 X 10-s in circular measure. 140 CHAPTER XI. ELECTRICITY. Electrqstatics, 163. If q denote the numerical value of a quantity of •electricity in electrostatic measure, the mutual force be- tween two equal quantities q at the mutual distance I will be^,. .In the C.G.S. system the electrostatic unit of electricity is accordingly that quantity which would repel an equal quantity at the distance of 1 centim. with a force of 1 dyne. Since the dimensions of force are — , we have, as regards dimensions, q- ml , o mP iA^~^ 72 = 72 ' whence ^ = —, q = m It . 164. The work done in raising a quantity of electricity -q through a difference of potential v is qv. Hence we have v = '!^ = mPt'Km-k-h = m¥t~\ In the C.G.S. system the unit difference of potential is CHAP. XI.] ELECTRICITY. 141 that difference through which a unit of electricity must be raised that the work done may be 1 erg. Or, we may define potential as the quotient of quantity of electricity by distance. This gives V = 7/i*A"^ . l-^ = m^^r\ as before. 165. In the O.G.S. system the unit of potential is the potential due to unit quantity at the distance of 1 centim. The capacity of a conductor is the quotient of the quantity of electricity with which it is charged by the potential which this charge poduces in it Hence we have capacity = ? = m^I^r^ , m''h~h = l. The same conclusion might have been deduced from the fact that the capacity of an isolated spherical con- ductor is equal (in numerical value) to its radius. The C.G.S. unit of capacity is the capacity of an isolated sphere of 1 centim. radius. 166. The numerical value of a ctirrent (or the strength of a current) is the quantity of electricity that passes in unit time. Hence the dimensions of current are ?; that is, m^l^t~'-. t' The O.G.S. unit of current is that current which con- veys the above defined unit of quantity in 1 second. 167. The dimensions of resistance can be deduced from Ohm's law, which asserts that the resistance of a wire is the quotient of the difference of potential of its two ends, by the current which passes through it. Hence we have resistance = m^l^t '^ ,m~H~ ^f = I'^t, \ 142 UNITS AND PHYSICAL CONSTANTS. [chap. Or, the resistance of a conductor is equal to the time required for the passage of a unit of electricity through it, when unit difference of potential is maintained between its ends. Hence resistance = ^5?i2ii:2^?H!!!^ = <. mi iir> .«»- ir ?«= rt. quantity 168. As the force upon a quantity q of electricity, in a field of electrical force of intensity i, is iq^ we have The quantity here denoted by i is commonly called the " electrical force at a point." Electromagnetics, 169. A current C (or a current of strength C) flowing along a circular arc, produces at the centre of the circle an intensity of magnetic field equal to C multiplied by length of arc divided by square of radius. Hence C divided by a length is equal to a field-intensity, or C = length X intensity = L . M^L " *T"' = l4m4t~\ 170. The quantiti/ of electricity Q conveyed by a cur- rent is the product of the current by the time that it lasts. The dimensions of Q are therefore L^M* 171. The work done in urging a quantity Q through a circuit, by an electromotive force E, is EQ ; and the work done in urging a quantity Q through a conductor, by means of a differeyice of potential E between its ends, is EQ. Hence the dimensions of electromotive force, and also the dimensions of potential, are ML-T"^ . L ~ ^M " ', or XI.] ELECTRICITY. 143 172. The capacity of a conductor is the quotient of quantity of electricity by potential. Its dimensions are therefore M*L* . M " ^L - 'P ; that is, L'^T*. 173. Resistance is — ; its dimensions are therefore C m4l^T-» . M " *L " *T ; that is, LT~\ 174. The following table exhibits the dimensions of each electrical element in the two systems, together with their ratios : — ! 1 Dimensions in electrostatic system. Dimensions in electromagnetic system. Dimensions in E.S. Dimensions in E.M. Quantity, M*L*T-i m4l* LT-i Current, M*L^T-2 M*L*T-i LT-i Capacity, L L-1T2 L2X-2 Potential and ) electromo- > tive force, ) U^Juh-^ mMt-2 , L-iT Resistance, L-iT LT-i L-2T2 175. The heat generated in time T by the passage of a current C through a wire of resistance R (when no other C^RT work is done by the current in the wire) is — = — gramme J degrees, J denoting 4*2 x 10^ ; and this is true whether C and R are expressed in electromagnetic or in electrostatic units. 144 UNITS AND PHYSICAL CONSTANTS. [chap. Ratios of the two sets of Electric Units, 176. Let us consider any general system of units based on a unit of length equal to L centims., a unit of mass equal to M grammes, a unit of time equal to T seconds. Then we shall have the electrostatic unit of quantity equal to M^L'T"! C.G.S. electrostatic units of quantity, and the electromagnetic unit of quantity equal to M'L^C.G.S. electromagnetic units of quantity. It is possible so to select L and T that the electrostatic unit of quantity shall be equal to the electromagnetic unit. We shall then have (dividing out by M^L^) LT~^ C.G.S. electrostatic units = 1 C.G.S. electromagnetic unit ; or the ratio of the C.G.S. electromagnetic unit to the C.G.S. electrostatic unit is — . Now — is clearly the value in centims. per second of that velocity which would be denoted by unity in the new system. This is a definite concrete velocity ; and its numerical value will always be equal to the ratio of the electromagnetic to the electrostatic unit of quantity, whatever units of length, mass, and time are employed. 177. It will be observed that the ratio of the two units of quantity is the inverse ratio of their dimensions ; and XI.] ELECTRICITY. 145 the same can be proved in the same way of the other four electrical elements. The last column of the above table shows that M does not enter into any of the ratios, and that L and T enter with equal and opposite indices, showing that all the ratios depend only on the velocity -. Thus, if the concrete velocity _ be a velocity of v centims. per second, the following relations will subsist between the C.G.S. units : — 1 electromagnetic unit of quantity = v electrostatic units. 1 „ „ current =1? „ 1 „ „ capacity =i;2 „ V electromagnetic units of potential = 1 electrostatic unit. v^ „ „ resistance = 1 „ 178. Weber and Kohlrausch, by an experimental comparison of the two units of quantity, determined the value of t; to be 3*1074 X 10^^ centims. per second. Sir. W. Thomson, by an experimental comparison of the two units of potential, determined the value of v to be 2-825 X IQio. Professor Clerk Maxwell, by an experiment in wliich an electrostatic attraction was balanced by an electro- dynamic repulsion, determined the value of v to be 2-88 X IQio. Professors Ayr ton and PeiTy, by measuring the capacity K 146 UNITS AND PHYSICAL CONSTANTS. [chap. of an air-condenser both electroraagnetically and statically ("Nature," Aug. 29, 1878, p. 470), obtained the value 2-98 X 1010. Professor J. J. Thomson (" Phil. Trans.," 188 3, June 21), by comparing the electrostatic and electromagnetic mea- sures of the capacity of a condenser, and employing Lord Rayleigh's latest value of the B.A. resistance coils, de- termined V to be 2-963x1010. All these values agree closely with the velocity of light in vacuo, of which the best determinations are, some of them a little less, and some a little greater than 3 X 1010. We shall adopt this round number as the value of v. 179. The dimensions of the electric units are rather simpler when expressed in terms of length, density, and time. Putting D for density, we have M = L^D. Making this substitution for M, in the expressions above obtained (§ 1 74), we have the following results : — Electrostatic. Electromagnetic. Quantity, D*L*r-i D^U Current, D^L^T-^ D*L^-i Capacity, L L-*T* Potential, D^L^T"' D^L^T-' Resistance, L-^T LT-i It will be noted that the exponents of L and T in these expressions are free from fractions. XI.] ELECTRICITY. 147 Specific Inductive Capacity, 180. The specific inductive capacity of an insulating substance is the ratio of the capacity of a condenser in which this substance is the dielectric to that of a conden- ser in other respects equal and similar in which air is the dielectric. It is of zero dimensions, and its value exceeils unity for all solid and liquid insulators. According to MaxwelPs electro-magnetic theory of light, the square root of the specific inductive capacity is equal to the index of refraction for the rays of longest wave- length. Messrs. Gibson and Barclay, by experiments performed in Sir W. Thomson's laboratory ("Phil. Trans.," 1871, p. 573), determined the specific inductive capacity of solid paraffin to be 1*977. Dr. J. Hopkinson (*' Phil. Trans.," 1877, p. 23) gives the following results of his experiments on difierent kinds of flint glass : — Kind of Flint Glass. Very light» ... Ligbt, Dense, Double extra ) dense, \ Specific Inductive Quotient Index of Density. by Refraction Capacity. Density. for D line. 2-87 6-57 2-29 1-641 3-2 6-85 214 1-574 3-66 7-4 202 1-622 4-5 101 2-25 1-710 1 1 In a later series of experiments (" Phil. Trans.," 1881, Dec. 16), Dr. Hopkinson obtains the following mean determinations : — 148 UNITS AND PHYSICAL CONSTANTS. [chap. Specific Specific Glass. Inductive Density. Inductive Cai)acity. Capacity. Hard crown, 6*96 2*485 Paraflfin, 2*29 A^ery light flint, 6*61 2*87 Light flint, 672 3*2 Dense flint, 7*38 3*66 Double extra-dense flint, 9*90 4*5 Plate, 8-45 181. For liquids Dr. Hopkinson ("Proc. Roy. Soc," Jan. 27, 1881) gives the following values of /x^ (computed) and K (observed), K denoting the specific inductive capacity and /x^ the index of refraction for very long waves deduced by the formula where 6 is a constant. Petroleum spirit (Field's), 1 *922 1 -92 Petroleum oil (Field's), 2*075 2*07 ,, (common), 2*078 2*10 Ozokerit lubricating oil (Field's) 2086 2*13 Turpentine (commercial), 2*128 2*23 Castor oil, 2*153 4*78 Sperm oil, 2*135 3*02 Olive oil, 2*131 3*16 Neatsfoot oil, 2*125 3*07 This list shows that the equality of /tx^ to K (which Maxwell's theory requires) holds nearly true for hydro- carbons, but not for animal and vegetable oils. 182. Wiillner (" Sitzungsber. konigl. bayer. Akad.," March, 1877) finds the following values of specific inductive capacity : — Paraffin, 1 *96 Shellac, ... 2 95 to 3 *73 Ebonite 2*56 Sulphur, ... 2*88 to 3*21 Plate glass, ... 6*10 XI.] ELECTRICITY. 149 Boltzmann ("Carl's Repertorium," x. 92—165) finds the following values : — Paraffin, 2*32 Colophonium, ... 2*55 Ebonite, 3*16 Sulphur, 3*84 SchUler ("Pogg. Ann.," clii. 535, 1874) finds:— Paraffin, ... 5 83 to 2*47 Caoutchouc, ... 2*12 to 2*34 Ebonite, ... 2*21 to 2*76 Do., vulcanized, 2 '69 to 2*94 Plate glass, 5*83 to 6*34 Silow (" Pogg. Ann.," clvi. and clviii.) finds the following values for liquids : — Oil of turpentine, 2*155 to 2*221 Benzene, 2*199 Petroleum, 2*039 to 2*071 Boltzmann (" Wien. Akad. Ber." (2), Ixx. 342, 1874) finds for sulphur in directions parallel to the three princi- pal axes, the values 4*773, 3-970, 3-811. 183. Quincke (" Sitz. Preuss. Akad.," Berlin, 1883j has made the following determinations. To explain the last two columns it is to be observed that, according to Max- well's theory, the charging of a condenser produces tension (or diminution of pressure) in the dielectric along the lines of force, and repulsion (or increase of pressure) perpen- dicular to the lines of force, the tension and the repulsion being each equal to K (A-B)2 87rc2 where K denotes the specific inductive capacity of the dielectric, c the distance between the two parallel plates of the condenser, and A - B their difference of potentials. Quincke observed the tension and repulsion, and computed 150 UNITS AND PHYSICAL CONSTANTS. [chap. K from each of them separately. The results are given in the last two columns, and are in every case greater than the " observed " value of K obtained in the usual way by comparison of capacities. The temperature printed below the index of refraction is the temperature at which the electrical experiments were performed. Ether, »i 5 vols, ether to 1 bisul- phide of carbon, 1 ether to 1 bisulphide, ... 1 ether to 3 bisulphide, ... Sulphur in bisulphide of carbon (19 5 per cent.) Bisulphide of carbon from Kahlbaum, Bisulphide of carbon from Heidelberg, 1 vol. bisulphide to 1 tur- pentine, Heavy benzol from ben- zoic acid, Pure benzol from benzoic acid, Light benzol, Rape oil, Oil of turpentine, Rock oil, Density tempera- tnre. 1-7205 atl4°-9 •8134 16° -4 •9966 16'* -6 M360 17"-4 1 -3623 12''-6 1 -2760 12° -2 1 ^2796 10° -2 1-0620 17°-8 •8825 15° -91 •8822 17°-64 •7994 17°^20 •9159 16°^4 •8645 17'1 •8028 17°0 Index of refnetion and tem- perature. 13605 6° -60 1-3594 8° -37 1-4044 8° -50 1-4955 10°-50 15677 5°30 1 6797 8°^68 1 0386 7" 50 1 -6342 12°^98 1 ^5442 10°-92 1 -5035 13° -20 1 •SOSO 14° -40 1^4535 11° -60 1 4743 16°-41 1 -4695 16° -71 1^4483 16°^62 Computed T Specific indnctlre /^ capacity from from obMired. tension. xepalBion . 3-364 4-851 4672 3-322 4623 4-660 2871 4-136 4-392 2-458 3 539 3 392 2-396 3-132 3061 2113 2-870 2-895 2-217 2-669 2743 1-970 2692 2^752 1-962 2-453 2 540 1-928 2-389 2^370 2050 2325 2375 r775 2-155 2-172 2-443 2-385 3296 1-940 2-259 2-356 1-705 2138 2-149 XI.] ELECTRICITY. 151 184. Professors Ayrton and Perry have found the following values of the specific inductive capacities of gases, air being taken as the standard : — Air, 1-0000 Vacuum, 0*9985 Carbonic acid, ... 1 -0008 Hydrogen, 0*9998 Coal gas, 1-0004 Sulphurous acid, 1 -0037 Practical Units, 185. The unit of resistance chiefly employed by practical electricians is the Ohrriy which is theoi*etically defined as 10^ C.G.S. electro-magnetic units of resistance. The practical unit of electro-motive force is the Volty ' which is defined as 10® C.G.S. electro-magnetic units of potential. The practical unit of current is the Ampere. It is de- fined as tV of the C.G.S. electro-magnetic unit current, or as the current produced by 1 volt through 1 ohm. The practical unit of quantity of electricity is the Coulomb, It is defined as tV of the C.G.S. electro-magnetic unit of quantity, or as the quantity conveyed by 1 ampere in 1 second. The practical unit of capacity is the Farad.* It is defined as 10~® of the C.G.S. electro-magnetic unit of capacity, or as the capacity of a condenser which holds 1 coulomb when charged to 1 volt. * As the farad is much too large for practical convenience, its millionth part, called the microfarad, is practically employed, and condensers are in use having capacities of a microfarad and its decimal subdivisions. The microfarad is lO'^'^ of the C.G.S. electromagnetic unit of capacity. 152 UNITS AND PHYSICAL CONSTANTS. [chap. The practical unit of work employed in connection with these is the Joule. It is defined as 10^ ergs, or as the work done in 1 second by a current of 1 ampere in flowing through a resistance of 1 ohm. The corresponding practical unit of rate of working is the Watt. It is defined as 10^ ergs per second, or as the rate at which work is done by 1 ampere flowing through 1 ohm. 186. The standard resistance-coils originally issued in 1865 as representing what is now called the ohm, were constructed under the direction of a Committee of the British Association, and their resistance was generally called the B. A. unit. The latest and best determinations by Lord Rayleigh and others have shown that it was about 1 — or, more exactly, 13 per cent. — too small, the actual resistance of the original B. A. coils being •987 X 109 c.G.S. 187. An earlier unit in use among electricians was Siemens* unit, defined as the resistance at 0" C. of a column of pure mercury 1 metre long and 1 sq. millimetre in section. The resistance of such a column is about •943 X 109 c.G.S. The reciprocal of -943 is 1 06. 188. The question of what electrical units should be adopted received great attention at the International Congress of Electricians at Paris in 1881 ; and the follow- ing resolutions were adopted : — XI.] ELECTRICITY. 153 Resolutions adopted by the International Congress of Electricians at the sitting of Septemnher 22nd, 1881. 1. For electrical measurements, the fundamental units, the centimetre (for length), the gramme (for mass), and the second (for time), are adopted. 2. The ohm and the volt (for practical measures of resistance and electromotive force or potential) are to keep their existing definitions, 10^ for the ohm, and 10^ for the volt. 3. The ohm is to be represented by a column of mercury of a square millimetre section at the temperature of zero centigrade. 4. An International Commission is to be appointed to determine, for practical purposes, by fresh experiments, the length of a column of mercury of a square millimetre section which is to represent the ohm. 5. The current produced by a volt through an ohm is to be called an ampere. 6. The quantity of electricity given by an ampere in a second is to be called a coulomb. 7. The ca{)acity defined by the condition that a coulomb charges it to the potential of a volt is to be called a farad. 189. At a subsequent International Conference at Paris in 1884, it was agreed to define the ^^ legal ohm^^ as " the resistance of a column of mercury 106 centimetres long and 1 sq, millimetre in section, at the temperature of meltirig ice" The following summary of experimental results was laid before this Conference. The two columns of numerical values are inversely proportional, their common product being 100. One of them gives the value of Siemens' 154 UNITS AND PHYSICAL CONSTANTS. [chap. unit in terms of the theoretical ohm (10® C.O.S.), and the other gives the length of a column of pure mercury at 0*" C, 1 square millimetre in section, which has a resist- ance of 1 theoretical ohm. Year. Obeerver. Siemenfl' Unit in Ohms. Colnmn of Mercury cm. Method. 1864. British Assoc. Com., -9539 104 83 Brit. Association. 1881. Rayleigh & Shuster, •9436 105-98 Do. 1882. Rayleiffh, •9410 106-28 Do. 1882. H.Weber, -9421 10614 Do. 1874. Kohlrausch, •9442 105-91 Weber( 1 st method). 1884. Mascart, •9406 106 32 Do. 1884. Wiedemann, •9417 106-19 Do. 1878. Rowland, •9453 •9408 9406 105-79 106 30 106-32 Kirch hoff. 1882. Glazebrook, Do. 1884. Mascart, Do. 1884. F. Weber, •9400 105-37 Do. 1884. R6iti, •9443 105-90 Rditi. 1873. Lorenz, •9337 107-10 Lorenz. 1884. Loreuz, •9417 10619 Do. 1883. Rayleigh, •9412 10624 Do. 1884. Zenz, •9422 10613 Do. 1882. Doru, •9482 105 46 Weber (damping). 1883. Wild, •9462 105^68 Do. 1884. H. F. Weber, •9500 105-26 Do. 1866. .foule, •9413 •9430 106-23 106 04 Joule. Mean, Several of the most distinguished physicists present expressed their opinion that 106^2 or 106^25 centimetres was the most probable value of the required length ; but in order to obtain unanimity it was agreed to adopt the length 106 centimetres, as above stated. 190. By way of assisting the memory, it is useful to remark that the numerical value of the ohm is the same XI.] ELECTRICITY. 155 as the numerical value of a velocity of one earth-quadrant per second, since the length of a quadrant of the meridian is 10® centims. This equality will subsist whatever funda- mental units are employed, since the dimensions of resist- ance are the same as the dimensions of velocity. No special names have as yet been assigned to any electrostatic units. Electric SparL 191. Sir W. Thomson has observed the length of spark between two parallel conducting surfaces maintained at known ditferences of potential, and has computed the corresponding intensities of electric force by dividing (in each case) the difference of potential by the^ distance, since the variation of potential per unit distance measured in any direction is always equal to the intensity of the force in that direction. His results, as given on page 258 of " Papers on Electrostatics and Magnetism," form the first two columns of the following table : — DiBtance between Siirfaces. Intensity of force in Electrostatic Units. Difference of Potential between Surfaces. In Electrostatic Units. In Electromagnetic Units. •0086 •0127 •0127 •0190 •0281 •0408 •0663 •0584 •0688 •0904 •1056 •1325 267 1 257-0 262-2 224-2 200-6 151-5 1441 139-6 140-8 134-9 1321 131-0 2-30 3-26 3-33 4-26 5-64 6-18 8-11 8-15 9-69 12-20 13-95 17-36 6 90x1010 9-78 „ 9-99 ,. 12-78 „ 16-92 „ 18-54 „ 24-33 „ 2445 „ 29-07 „ 36 60 „ 41-85 „ £2-08 „ 156 UNITS AND PHYSICAL CONSTANTS. [chap. The numbers in the third column are the products of those in the first and second. The numbers in the fourth column are the products of those in the thii*d by 3 X 1010. 192. Dr. Warren De La Rue, and Dr. Hugo W. Miiller ("Phil. Trans.," 1877) have measured the striking dis- tance between the terminals of a battery of choride of silver cells, the number of cells being sometimes as great as 11000, and the electromotive force of each being 1*03 volt. Terminals of various forms were employed ; and the results obtained with parallel planes as terminals have been specially revised by Dr. De La Rue for the present work. These revised results (which were obtained by graphical projection of the actual observations on a larger scale than that employed for the Paper in the Philosophi- cal Transactions) are given below, together with the data from which they were deduced : — DATA. y No. of Cells. Striking Distance. In Inches. In Centims. 1200 0-012 0305 2400 •021 •0533 3600 •033 •0838 4800 •049 •1245 5880 •058 •1473 6960 •073 1854 8040 •088 •2236 9540 •110 •2794 11000 •133 •3378 XI.] ELECTRICITY. 157 DEDUCTIONS. Electromotive Force in Volts. Striking Distance in Gentims. Volts per Gentim. 1 Intensity of Force In C.O.S. units. 1 Electromagnetic. Electro- static. 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 11330 •0205 •0430 •0660 •0914 •1176 •1473 •1800 •2146 •2495 •2863 •3245 •3378 48770 46500 45450 43770 42510 40740 38890 37280 36070 34920 33900 33460 4-88 X 4-65 , 455 4-38 , 4-25 407 3-89 . 3-73 , 3-61 3-49 , 3 39 3-35 , 10^2 163 155 152 146 142 136 130 124 120 116 113 112 193. The resistance of a wire (or more generally of a prism or cylinder) of given material varies directly as its length, and inversely as its cross section. It is there- fore equal to ■^ length section' where R is a coefficient depending only on the material. R is called the specific resistance of the material. Its reciprocal — is called the specific conductivity of the R material. R is obviously the resistance between two opposite faces of a unit cube of the substance. Hence in the C.G.S. system it is the resistance between two opposite faces of a cubic centim. (supposed to have the form of a cube). The dimensions of specific resistance are resistance x length ; that is, in electromagnetic measure, velocity x length ; that is, L^T'i. 158 UNITS AND PHYSICAL CONSTANTS. [chap. Resistance. 194. The following table of specific resistances is altered from that given in former editions of this work by subtracting 1*88 per cent, from all the numbers in the column headed " Specific Resistance," this being the correc- tionrequired to reducetheresistanceof mercury from 96146, the value previously given, to 94340, which is the value resulting from the new definition of the " legalohm " : — Specific Resisla/aces in Electromagnetic Measure (ai 0° C, unless otherivise stated). 1 Specific Resistance. Percentage variation per degree at 20' C. Specific Gravity. Silver, hard-drawn 1579 1611 2114 19474 94340 10781 5-9x1018 7-05xlO'o 4^39 „ 3-26 „ 1-41 „ 1-24 „ 1-34 „ 1-83 „ 1-91 „ 2-23x1016 1 -36 X 10^5 1-45x101* 7-21x10" 346 X 1023 6-87 X 102* •377 •388 •365 •387 •072' •065 100 •47 •47 •653 •799 r259 1-410 10-50 895 19 27 11-391 13-595 15-218 CoDPer, ,, Gold, ,, Lead, pressed, Mercurv. liouid Gold 2, Silver 1, hard or]^ annealed, / Selenium at 100* C, crys-^ talline, / Water at 22*' C, „ with -2 per cent. H2SO4 >» >» 0*0 ,, ,) 20 li 91 "" »» >» >> »» '*■*■ »> >» Sulphate of Zinc and Water \ ZnS04 + 23 H2O at 23" 0. , j Sulph.of Copper and VVater\ CUSO4 + 45 flaO at 22' C, / Glass at 200'' C, 250' t1 «»w , 300^ ,, t^w , 400" Gutta Percha aiik'^C.]....'. o-c, XI.] ELECTRICITY. 159 For the authorities on which this table is based see Maxwell, "Electricity and Magnetism," vol. i., last chapter. 195. The following table of specific resistances of metals at 0° C. is reduced from Table IX. in Jenkin's Cantor Lectures. It is based on Matthiessen*s experi- ments. A deduction of 1 •SS per cent, has been made, as in the preceding table : — • Specific Resistance. Percentage of Variation for a degree at 20'* C. Resistance in Ohms of a Wire of 1 mm. diam. 1 m. long. Silver, annealed, „ hard-drawn, CoDDer. annealed 1492 1620 1584 1620 2041 2077 2889 5581 8982 9638 12358 13103 19468 35209 130098 94340 2419 20763 10779 •377 •388 •365 •365 •365 •387 •389 •354 •072 •031 •044 •065 •0190 •0206 •0202 1 ,, hard-drawn, Gk>ld, annealed, ., hard -drawn •0206 1 •0260 •0264 ' Aliuninium, annealed Zinc, pressed, Platinum, annealed •0368 1 •0749 i •1144 i Iron, annealed, Nickel, annealed, Tin, pressed, Lead, pressed, Antimon v« nressed •1227 1 •1573 , •1668 •2479 •4483 Bismuth, pressed, Mercury, liquid, Alloy, 2 parts Platinum, 1 ] paurt Silver, by weight, j- hard or annealed, j German Silver, hard or an-"^ nealed, / Alloy, 2 parts Gold, 1 Sil- \ ver, by weight, hard or - annealed, 1 6565 1 r2012 ' •0308 •2644 •1372 160 UNITS AND PHYSICAL CONSTANTS. [chap. Resistances of Condtictors of Telegraphic Cables per nautical mile, at 24° C, in CG.S. units. Red Sea, 7*79xl09 Malta- Alexandria, mean, 3'42 Persian Gulf, mean, 6*17 Second Atlantic, mean, 4*19 196. The following formulse are given by Benoit* for the ratio of the specific resistance at i° C. to that at 0° C. : — Aluminium, 1+003876^ 4-*00000l320^ Copper, l4--00367^ 4- -000 000 687^2 Iron, 14--004516< + '000 006 828^2 Magnesium, 1 + '003870^ + '000 000 863^2 Silver, l + *003972< +'000000687^ Tin, l + -004028« + 000 005 826«* Mercury in glass tube, ' apparent resistance, not corrected for expansion, ^ 1 + -0008649^ + -000 001 12^^ Adopting the formula 1+at for the ratio of the specific resistance at t" to that at 0**, MM. Cailletet and Bouty ("Jour, de Phys.," July, 1885) have made the following determinations of the coefficient of variation a at very low temperatures : — Aluminium, Copper, Iron, Magnesium, Mercury, Silver, +30 to-102 '00385 Tin, to- 85 '00424 The new alloy called platinoid (consisting of German silver with a little tungsten) has been fouiid by Mr. J. T. * Benoit, *' Etudes exp^rimentales sur la Resistance ^lectrique sous rinfluence de la Temperature." Paris, 1873. Range of Temperature. CoefiScient of Variation. + 28" to- 91" .. -00385 -23 to-123 ... -00423 to- 92 .. -0049 to- 88 ... -00390 -40 to- 92 .. -00407 XI.] ELECTRICITY. 161 Bottoraley ("Proc. Roy. Soc.," May 7, 1885) to have an average variation of resistance with temperature of only •022 per cent, per degree centigrade, between 0° C. and 100° C, being about half the variation of German silver. Its specific resistance ranges in different specimens from 2-9 X 10-*^ to 3-7 X 10-» C.aS. Resistances of Liquids, 197. The following tables of specific resistances of solutions are from the experiments of Ewing and Macgregor ("Trans. Roy. Soc., Edin.," xxvii. 1873) :— Solutions at 10° C. Specific Resistance. Sulphate of Zinc, saturated, 3 '37 x 10^® ,, „ minimum, 2*83 Sulphate of Copper, saturated, 2 '93 Sulphate of Potash, „ 1*66 Bichromate of Potash, ,, 2*96 The following table is for solutions of sulphate of cop- per of various strengths. The first column gives the ratio by weight of the crystals to the water in which they are dissolved : — strength, ^t^^'a »» )) »> >> 1 to40 30 20 10 7 5 10167 10216 1 0318 10622 1 -0858 1-1174 Specific Resistance. 16-44x1010 13-48 9-87 5-90 4-73 3-81 »> Strength. Density. 1 to 4-146 1 4 1 3-297 1 3 1 2-5971 1 saturated/ 1386 1432 1679 1823 Specific Resistance. 3-5 xlO^'' 3-41 3-17 3 06 >» J > 2051 2-93 >» »> The following table is for solutions of sulphate of zinc : — strength. Density. Ito40 20 10 7 5 10140 1-0278 10540 1 0760 11019 Specific Resistance. 18-29x1010 11-11 6-38 5-08 4-21 9) »> >' »5 Strength. Density at 10°. lto3 1-1582 2 1-2186 1 -5 1 -270 1 1 -3530 saturated/ Specific Resistance. 3-37x1010 3 03 2-85 310 »> )} 162 UNITS AND PHYSICAL CONSTANTS. [chap. The following table for dilute sulphuric acid is from Becker's experiments, as quoted by Jamin and Bouty, torn. iv. p. Ill : — Specific Resistance Density. 110 1-20 1-25 1-30 1-40 1-50 1-60 1-70 /AtO\ 1-37x1 010 1 33 1-31 1-36 1-69 2-74 4-82 9-41 At 8°. 104 xlO'® •926 •896 •94 r30 213 3 62 )) *» »» 5» »» S» At 16'. •845 X 10^0 •666 •624 •662 105 1-72 6-25 „ 2-75 423 it »» )) At 24'. ^ •737 X 10^0 •486 •434 •472 •896 1-52 2-21 307 »» >» >» » Resistance of Carbons, 198. The specific resistance of Carry's electric-light carbons at 20" C. is stated to be 3-927 X 106 C.G.S., whence it follows that the resistance of a cylinder 1 metre long and 1 centimetre in diameter is just half an ohm. The specific resistance of Gau din's carbons is about 8*5 x ID** n )) retort carbon j> 6-7 x 107 )» J) graphite from 2^4 x 1 0^ to 4^2 x 10^ The resistance of carbon diminishes as the temperature J_ 19 increases, the diminution from 0° to 100° C. being -^^ for Carry's and — for Gaudin's. The resistance of an incan- descent lamp when heated as in actual use is about half its resistance cold. XI.] ELECTRICITY. 163 Resistance of the Electric Arc, 199. The difference of potentials between the two carbons of an arc lamp has been found by Ayrton and Perry ("PhU. Mag.," May, 1883) to be practically in- dependent of the strength of the current, when the dis- tance between them is kept constant. It was scarcely altered by tripling the strength of the current. The apparent resistance of the arc (including the effect of reverse electromotive force) is therefore inversely as the current. The difference of potentials was about 30 volts when the current was from 6 to 12 amperes. 200. The following approximate determinations of the resistance of water and ice at different temperatures are contained in a paper by Professors Ayrton and Perry, dated March, 1877 (" Proc. Phys. Soc, London," vol. ii. p. 178):— Temp. Specific Cent. Resistance. o -12-4 2-240x1018 - 6*2 1-023 „ - 5 02 9-486x1017 - 3-5 6-428 „ - 3-0 5-693 „ 2-46 4-844 „ - 1-5 3-876 „ - 0-2 2-840 „ + 0-75 1-188 „ about + 2-2 2-48 x lO^e + 4-0 9-1 xlO^s + 7-75 5-4 xlO^^ + 11-02 3-4 „ The values in the original are given in megohms, and we have assumed the megohm = 10^^ C.G.S. units. 164 UNITS AND PHYSICAL CONSTANTS. [chap. According to F. Kohlrauscli (" Wied. Ann.," xxiv. p. 48, 1885) the resistance at 18° C. of water purified by distillation in vacuo is 4 x 10^^ times that of mercury. This makes its specific resistance 3-76 X 1015. 201. The specific resistance of glass of various kinds at various temperatures has been determined by Mr. Thomas Gray (" Proc Roy. Soc.," Jan. 12, 1882). The following are specimens of the results : — Bohemian Glass Tubing, density 2*43. At 60' 605x10^ At 160° 2*4 xW^ 100 2 xlO^i 174 8-7 xlO^s 130 2 X 1020 Thomson's Electrometer Jar (flint glass), density 3*172. At 100" 2-06 X 1023 At 160** 2*45 x lO^i 120 4-68x1022 180 5-6 x 102« 140 1-06 „ 200 1-2 ,, *-\/V J. *. ,, The following are all at 60" C. : — Bohemian Beaker, 4-25x1022 density 2-427 »» 19 7-15 „ 2-587 Florence Flask, 4-69 X 1020 2-523 Test Tube, 1-44 „ 2-435 j» »> 3-50 „ 2-44 Flint Glass Tube, 3-89x1022 2-753 Thomson's Electro- ' meter Jar (flint - 1-02x1024 3-172 glass). 202. The following appoximate values of the specific resistance of insulators after several minutes' electrifi- cation are given in a paper by Professors Ayrton and Perry ("Proc. Royal Society," March 21, 1878), "On the Viscosity of Dielectrics " : — XI.] ELECTRICITY. 165 Mica, 8*4xl(F 20 Ayrton and Perry. Gutta-Percha, 4-5xl0» 24 {«*^trmer1)ff '^ Shellac, O'OxlO^* 28 Ayrton and Perry. Hooper's Material, 1 *5 x lO^'^ 24 Recent cable tests. Ebonite, 2*8x1025 45 Ayrton and Perry. Paraffin, 3-4x102* 46 „ p J / Not yet measured with accuracy, but greater '** \ than any of tbe above. Air, Practically infinite. 203. Particulars of Board of Trade Standard Gauge of Wires {Imperial Gauge) Nos. 4 to 20. Diameter. Resistance in ohms of 1 metre Sectional area. Sq. inchea length pure c opper at 0° C. No. MiUi- metres. Thou- sandths of inch. 1 Annealed. Hard-drawn. 1 4 5-89 232 •04227 •0005929 •0006065 5 6-38 212 •03530 7107 7269 6 4-88 192 •02895 8638 8835 7 4-47 176 •02433 •001029 •001053 8 406 160 •02011 1248 1276 9 3-66 144 •01629 1536 1571 10 3-25 128 •012S7 1948 1992 11 2-95 116 -01057 2364 2418 12 2-64 104 •008494 2951 3019 13 2-34 92 •006647 3757 3842 14 203 80 •005026 4992 5106 15 1-83 72 •004070 6142 6283 16 1-63 64 •003216 7742 7919 17 1-42 56 •002463 •01020 ; ^01043 18 1-22 48 •001809 •01382 1 -01414 19 1016 40 •001256 •01993 -02038 20 0-914 36 •000917 •02462 •02518 166 UNITS AND PHYSICAL CONSTANTS. Tchap. The heat generated per second in 1 metre length of — \ gm. deg.^ and (Q\2 — 1 gm. deg., C. denoting the current in amperes, and D the diameter in millimetres. 204. Resistance of 1 metre length of Wires of Imperial Gatige at 0** C, (For copper see preceding table.) Oerman Silver, No. Iron, annealed. an ha either nealed or rd*drawn. Platinum, annealed. Silver, annealed. 1 4 •00.3606 •007768 •003361 •0005583 5 4322 9311 4028 6692 6 5253 "01132 4896 8184 7 6261 01349 5836 9694 1 8 7590 01635 7074 •001175 ! i 9 9339 02012 8705 1446 1 ! 10 •01184 02552 •01104 1834 ; 11 •01438 1 •03096 •01340 2226 12 •01795 03867 •01673 2779 I 13 •02-285 04922 •02129 3538 ' 14 •03036 06540 •02829 4700 , 15 •03736 1 •08047 , •03482 5784 ■ 16 •04708 1 • 1014 •04388 7290 ; 17 •06204 1336 •05782 9606 18 •08405 1811 •07834 •01301 19 •1212 2611 •1130 •01876 20 , 1 •1498 3226 •1396 •02319 Electromotive Force. 205. The electromotive force of a Daniel I's cell was found by Sir W. Thomson (p. 245 of "Papers on Electricity and Magnetism ") to be •00374 electrostatic unit, XI.] ELECTRICITY. 167 from observation of the attraction between two parallel discs connected with the opposite poles of a Danieirs battery. As 1 electrostatic unit is 3 x 10^^ electromag- netic units, this is -00374 x 3 x 10io= M22 x lO® electro- magnetic units, or 1*122 volt. According to Latimer Clark's experimental determina- tions communicated to the Society of Telegi-aph Engineers in January, 1873, the electromotive force of a DanielUs cell with pure metals and saturated solutions, at 64° F., is 1*105 volt, and the electromotive force of a Grove's cell 1 '97 volt. These must be diminished by 1 per cent, because they were deduced from the assumption that the B. A. unit of resistance was correct. They will thus be reduced to 1*094 and 1*95 volts. According to the determination of F. Kohlrausch ("Pogg. Ann.," vol. cxli. [1870], and Erganz., vol. vi. [1874], p. 35) the electromotive force of a DanielFs cell is 1*138x108, and that of a Grove's cell 1*942x108. These must be diminished by 3 per cent., because they were deduced from the value '9717 x 10^ for Siemens' unit which is 3 per cent, too great. They will thus be reduced to 1*104 and 1*884 volts. H. S. Carhart ("Amer. Jour. Sci. Art.," Nov. 1884) has found the following different values for the electro- motive force of a Daniell's cell according to the strength of the zinc sulphate solution : — Per cent, of ZnSO^. Electromotive force in volts. Per cent. Electromotive force. 1 1*125 10 1*118 3 1*133 15 1*115 5 1*142 20 1111 74 1*1*20 25 1*111 168 UNITS AND PHYSICAL CONSTANTS. [chap. He finds by the same method the electromotive force of Latimer Clark's standard cell to be 1434 volt. LordRayleigh ("Phil. Trans.," June 1884, p. 452) has determined the electromotive force of a Clark cell at 15° C. to be 1 -435 volt. The value formerly assigned to it was 1*457 volt, and was based on the assumption that the B. A. unit of resist- ance was correct. In a supplementary paper (Jan. 21, 1886) he gives the general result for any temperature T, 1435{l-000077(«-15)}, together with full particulars as to the precautions neces- sary for securing constancy. 206. Professors Ayrton and Perry have made deter- minations of the electromotive forces called out by the contacts, two and two, of a great number of substances measured inductively. The method of experimenting is described in the Proceedings of the Royal Society for March 21, 1878. The following abstract of their latest results was specially prepared for this work by Professor Ayrton in January, 1879 : — ELECTBICITY. JO ami] ai(i }« sjo^sjailuiax sSusAy i i i i i I 3 ^ P I ? i — 3,1= = I11l Ivt ^ III! ffl sill I iillH ili's' i ! iili i ^ i iflill CONTACT DIFFERENCES OP POTENTIAL IN VOLTS. J 1 5 1 1 i P oi m™-'?. Dtatinedwatar, ..,.- Alum, saturated at: -Mi -■oas em -■m -108 -070 --173 ■Mi --006 ■171 -■m -720 to -ZS6 '34G --aw ■OH i-eoo ■872 ■177 'A'S'.T»"'";} Sea Hit, Bpeclflc ' 3 i -i-l H\ rated at 1&''&C., .,( Zlno lulphate lolu- lDl»t11Icdwntern,iiedl M DtotiUed -water, 1 stroug: HulphuMc 10 DlBflil^ -«aier. 6 Distilled -B3t6r, strong Bulphurlo e BtroDg eulphurle -■» •^SSlii Sulphuric ttcfd Kitricacid MercurouB Bulphate Distilled water, »itb ] atn^eofsmpliurio T^ average tempeni All tbe JJqulda and HlU employed n ff, were only coinmercliiUs pure. Solids fcith Liquids and Lvjuids with Liquids in Air, • a 1 o Amalgamated Zinc. n • 1 •2 1 AlumSolution.satu- rated at 16°^6 C. Copper Sulphate Solution, satu- rated at 16° C. Zinc Sulphate Solu- tion, Specific Grav- ity 1^125 at 16° '90. Zinc Sulphate Solu- tion, saturated at 15°^3 C. 1 Distilled Water, 3 Zinc Sulphate. Strong Nitric Acid. -•105 to +•156 •231 • • • • • • -•043 ■ • •164 1 -•686 • ■ -•014 • • • • • a • • • • • • • t •090 • • • • • • • • -•043 ■ • • • ■ • •095 •102 -•666 • • -•436 . -•637 -•348 1 -•288 • -•430 -•284 • • • • -•200 -•095 -•444 ■ • • • • • • • • • - -102 -•344 ■ • -•868 • • -•429 • • •016 ■ « •848 ■ • • • 1^298 1^466 1-269 1^699 • • • • • • •476 -•241 • • • • • • • • t • • • • • •078 Example of the above table :— Lead is poaitivo to d\&\iiil\&^'««.\x:;c^«s^^ the ooatact difference o/ potentials is O^lTl volt. 172 UNITS AND PHYSICAL CONSTANTS. [chap. The authors point out that in all these experiments the unknown electromotive forces of cei*tain air contacts are included. From these tables we find we can build up the electro- motive forces of some well-known cells. For example, in a DanielFs cell there are four contact differences of potential to consider, and in a Grove's cell five, viz. : — DanielVa Cell. Volte. Copper and saturated copper sulphate, +0*070 Saturated copper sulphate and saturated zinc sulphate, - 0*095 Saturated zinc sulphate and zinc, +0*430 Zinc and copper, +0*750 M55 Grovels Cell. Copper and platinum, +0*238 Platinum and strong nitric acid, +0*672 Strong nitric acid and very weak sulphuric acid, +0*078 Very weak sulphuric acid and zinc, +0'241 Zinc and copper, +0*750 1*979 TJiermoelectricity, 207. The electromotive force of a thermoelectric circuit is called Thermoelectric force. It is proportional ccet. par. to the number of couples. The thermoelectric force of a single couple is in the majority of cases equal to the product of two factors, one being the difference of temperature of the two junctions, and the other the difference of the thermo- electric heights of the two metals at a temperature midway between those of the junctions. The current through the hot junction is from the lower to the higher metal when their heights are measured at the mean temperature. XI.] ELECTRICITY. 173 Our convention as to sign (that is, as to up and down in speaking of thermoelectric height) is the same as that adopted by Prof. Tait, and is opposite to that adopted in the first edition of this work. We have adopted it because it leads to the rule (for the Peltier and Thomson effects) that a current running down generates heat, and a current running up consumes heat. The following table of thermoelectric heights relative to lead can be employed when the mean temperature of the two junctions does not differ much from 19° or 20° 0. It is taken from Jenkin's " Electricity and Magnetism,'*^ p. 176, where it is described as being compiled from Matthiessen's experiments. We have reversed the signs to suit the above convention, and have multiplied by 100 to reduce from microvolts to O.G.S. units. l^hermoelectric Heights at about 20° C. Bismuth, pressed comO _ q-qq mercial wire, / Bismuth, pure pressed \ _qqqq wire, / Bismuth, crystal, axial, - 6500 , , equatorial ,....— 4500 Cobalt, -2200 German Silver, -1175 - 41-8 + 10 + 10 + 90 + 120 Quicksilver, Lead, Tin, Copper of Commerce, Platinum, Gold, Antimony, pressed wire + Silver, pure hard, + Zinc, pure pressed, + Copper, galvano-plas- \ 280 300 370 380 600 tically precipitated, j Antimony, pressed \ commercial wire, ...) Arsenic, + 1356 Iron, pianoforte wire, + 1750 Antimony, axial, + 2260 ,, equatorial, + 2640 Phosphorus, red, + 2970 Tellurium, +50200 Selenium, + 8070O 208. The following table is based upon Professor TaiVs thermoelectric diagram (" Trans. Roy. Soc, Edin.," vol. xxvii. 1873) joined with the assumption that a Grove's cell has electromotive force 1*97 x 10^ : — »» 5? 174 UNITS AND PHYSICAL CONSTANTS. [chap. Thermoelectric Heifirhts at <• C. in C.G.S. units. Iron, + 1734- 4*87 « Steel, + 1139- 3*28^ Alloy, believed to be Platinum Iridium, + 839 at all temperatures. Alloy, Platinum 95; Iridium 5, + 622- '55 1 90; „ 10, + 696- l-34« 85; „ 15, + 709- "63^ M ,, 85; ,, 15, + 577 at all temperatures. Soft Platinum, - 61- 1*10^ Alloy, platinum and nickel, + 544- 1*10^ Hard Platinum, + 260- '75^ Magnesium, + 244- '95^ German Silver, _1207- 5-12^ Cadmium, + 266+ 4*29« Zinc, + 234+ 2-40« Silver, + 214+ l-50« Gold, + 283+ l'02t Copper, + 136+ '95t Lead, Tin, - 43+ '55t Aluminium, - 77+ '39^ Palladium, - 625- 3-59^ Nickel to HS^C, -2204- 5'\2t 250" to 310° C, -8449 + 24-U from 340° C, - 307- 5-12< The lower limit of temperature for the table is - 18° C. for all the metals in the list. The upper limit is 416° C, with the following exceptions : — Cadmium, 258° C; Zinc, 373'' C. ; German Silver, 175° C. Ex. 1. Required the electromotive force of a copper-iron couple, the temperatures of the junctions being 0* C. and 100° C. We have, for iron, + 1734 - 4-87< ; ,, copper, + 136+ -95^; „ iron above copper, 1598-5*82^. 5» J) XI. 1 ELECTRICITY. 175 The electromotive force per degree is 1598-5-82x50 = 1307, and the electromotive force of the couple is 1307(100-0) = 130,700, tendiog from copper to iron through the hot junction. By the neutral point of two metals is meant the tem- perature at which their thermoelectric heights are equal. Ex. 2. To find the neutral point of copper and iron we have 1598-5-82^ = 0, < = 275; that is, the neutral point is 275° 0. When the mean of the temperatures of the junctions is below this point, the current through the warmer junction is from copper to iron. The current ceases as the mean temperature attains the neutral point, and is reversed in passing it Ex. 3. F. Kohlrausch ('*Pogg. Ann. Erganz.," vol. vi. p. 35, 1874) states that, according to his determination, the electromotive force of a couple of iron and German silver is 24 X 10^ millimetre-milligramme-second units for 1° of difference of temperatures of the junctions, at moderate temperatures. Compare this result with the above Table at mean temperature 100". The dimensions of electromotive force are M^L'T"* ; hence the C.G.S. value of Kohlrausch's unit islO'^lO"' = 10"', giving 2400 as the electromotive force per degree of difference. From the above table we have Iron above German silver, 2941 + -25^, which, for t = 1 00, gives 2966 as the electromotive force per degree of difference. 176 UNITS AND PHYSICAL CONSTANTS. [chap. Peltier and Thomson Effects. 209. When a current is sent through a circuit com- posed of different metals, it produces in geneiul three distinct thermal effects. 1. A generation of heat to the amount per second of C^R ergs, denoting the current, and R the resistance. 2. A generation of heat or cold at the junctions. This is called the Peltier effect, and its amount per second in ergs at any one junction can be computed by multiplying the difference of thermoelectric heights at this junction by < + 273 and by the current, t denoting the centigrade temperature of the junction. If the current flows down (that is from greater to less thermoelectric height) the effect is a warming ; if it flows up, the effect is a cooling. Ex. 4. Let a unit current (or a current of 10 amperes) flow through a junction of copper and iron at 100* C. The thermoelectric heights at 100° C. are Iron, 1247 Copper, 231 Iron above copper, 1016 Multiplying 1016 by 373, we have about 379,000 ergs, or — - of a gramme-degree, as the Peltier effect per second. Heat of this amount will be generated if the current is from iron to copper, and will be destroyed if the current is from copper to iron. 3. A generation of heat or cold in portions of the cir- cuit consisting of a single metal in which tlie temperature varies from point to point. This is called the Thomson effect. Its amount per second, for any such portion of XI.] \ ELECTRICITY. 177 the circuity is the difference of the thermoelectric heights of the twc ends of the portion, multiplied by 273 + <, i«rhere t denotes the half-sum of the centigrade tempera- tures of the ends, and by the strength of the current. The Thomson effect, like the Peltier effect, is reversed by reversing the current, and follows the same rule that heat is generated when the current is from greater to less thermoelectric height. Experiment shows that the Thomson effect is insensible in the case of lead; hence the thermoelectric height of lead must be sensibly the same at all temperatures. It is for this reason that lead is adopted, by common consent, as the zero from which thermoelectric heights are to be reckoned. Ex. 5. In an iron wire with ends at 0° C. and 100° C, the cold end is the higher (thermoelectrically) by 4-87 X 100— that is, by 487. Multiplying this differ- ence by 273 + 1(0 + 100) or 323, we have 157300 as the Thomson effect per second for unit current. This amount of heat (in ergs) is generated in the iron when the current through it is from the cold to the hot end, and is destroyed when the current is from hot to cold. Ex. 6. In a copper wire with ends at 0° 0. and 100** 0., the hot end is the higher by '95 x 100 or 95. Multiply- ing this by 323, we have 30700 (ergs) as the Thomson effect per second per unit current. This amount of heat is generated in the copper when the current through it is from hot to cold, and destroyed when the current is from cold to hot. The effect of a current from hot to cold is opposite in these two metals, because the coefficients of ^ in the M / y 178 UNITS AND PHYSICAL CONSTANTS./ [chap. expressions for their thermoelectric heights (p. 174) have opposite signs. Relation between Thermoelectric Force and the Peltier and Thomson effects, 210. The algebraic sum of the Peltier and ThomsoD effects (expressed in ergs) due to unit current for one second in a closed metallic circuit, is equal to the thermoelectric force of the circuit; and the direction of this thermoelectric force is the direction of a current round the circuit which would give an excess of destruction over generation of heat (so far as these two effects are concerned). Ex. 7. In . a copper-iron couple with junctions at 0* C and 100* C, suppose a unit current to circulate in such a direction as to pass from copper to iron through the hot junction, and from iron to copper through the cold junction. The Peltier effect at the hot junction is a destruction of heat to the amount 1016 x 373 = 379,000 ergs. The Peltier effect at the cold junction is a generation of heat to the amount 1598 x 273 = 436,300 ergs. The Thomson effect in the iron is a destruction of heat to the amount 487 x 323 = 157,300 ergs. The Thomson effect in the copper is a destruction of heat to the amount 95 x 323 = 30,700 ergs. The total amount of destruction is 567,000, and of generation 436,300, giving upon the whole a destruction of 130,700 ergs. The electromotive force of the couple is therefore 130,700, and tends in the direction of the current here supposed. This agrees with the calculation in Example 1. " ^ ^ \ XI.] ELECTRICITY. 179 Electrochemical Equivalents, 211. The quantity of a given metal deposited in an electrolytic cell or dissolved in a battery cell (when there is no " local action ") depends on the quantity of electricity that passes, irrespective of the time occupied. Hence we can speak definitely of the quantity of the metal that is " equivalent to " a given quantity of electricity. By the electrochemical equivalent of a metal is meant the quantity of it that is equivalent to the unit quantity of electricity. In the C. G. S. system it is the number of grammes of the metal that are equivalent to the C.G.S. electromagnetic unit of electricity. Special attention has been paid to the electrochemical equivalent of silver, as this metal afibrds special facilities for accurate measurement of the deposit. The latest experiments of Lord Rayleigh and Kohlrausch agree in giving •01118 as the C.G.S. electrochemical equivalent of silver.* The number of grammes of silver deposited by 1 ampere in one hour is •01118 x^^^x 3600 = 4-025. 212. The electrochemical equivalents of the most im- ])ortant of the elements are given in the following table. They are calculated from the chemical equivalents in the l)receding column by simple proportion, taking as basis the above-named value for silver. Their reciprocals are the quantities of electricity required for depositing one * Rayleigh's detennination is 0111794; Kohlrausch's, '011183; Mascart's, -011156. See " PhU. Trans.," 1884, pp. 439, 458. 180 UNITS AND PHYSICAL CONSTANTS. [chap. gramme. The quantity of electricity required for deposit- ing the number of grammes stated in the column '' chemical equivalents" is the same for all the elements, namely, 9634 C.G.S. units. Elemouts. Mectro-positive — Hydrogen, Potassium, Sodium, Gold, Silver, Copper (cupric), ,, (cuprous), Mercury (mercuric),.. ,, (mercurous), Tin (stannic), ,, (stannous), Iron (ferric), „ (ferrous), Nickel, Zinc, Lea<l, Aluminium, Electro-negative — Oxygen, Chlorine, Iodine, , Bromine, , Nitrogen, Atomic Weight. • a I 1 1 39-03 1 23 00 1 196-2 3 107-7 1 63-18 2 199-8 1 2 It 117-4 1 4 55-88 2 3 58-6 2 2 64-88 2 206-4 2 27-04 3 15-96 2 35-37 1 126-54 1 79-76 1 14-01 3 Chemi- cal Equiva- lents. 1 39-03 23-00 65-4 107-7 31-59 63-18 99-9 199-8 29-35 58-7 18-63 27-94 29-3 32-44 103-2 901 7-98 35-37 126-54 79-76 4-67 Electro- chemical equivalents orgrammes per unit of electricity. •0001038 •004051 •002387 •006789. •01118 •003279 •006558 •01037 -02074 -003046 •006093 -001934 -002900 -003042 -003367 •01071 -000935 -0008283 -003671 •013134 •008279 -0004847 Recipro- cal or Electri- city per gramme. 9634 246-9 418-9 147-3 89-45 305-0 152-5 96-43 48-22 328-3 164-1 517-1 344-8 328-7 297-0 93-37 1070 1207 272-4 76-14 120-8 2063 To find the equivalent of 1 coulomb, divide the above electrochemical equivalents by 10. To find the number of grammes deposited per hour by 1 ampere, multiply the above electrochemical equivalents by 360. XI.] ELECTEICITY. 181 213. Let the " chemical equivalents " in the above table be taken as so many grammes ; then, if we denote by H the amount of heat due to the whole chemical action which takes place in a battery cell during the consumption of one equivalent of zinc, the chemical energy which runs down, namely JH ergs, must be equal (if there is no wasteful local action) to the energy of the current pro- duced. But this is the product of the quantity of electricity 9634 by the electromotive force of the cell. TTT The electromotive force is therefore equal to -zrzr-* ^ 9634 In the tables of heats of combination which are in use among chemists, the equivalent of hydrogen is taken as 2 grammes, and that of zinc as 64*88 or 65 grammes. The equivalent quantity of electricity will accordingly be 9634 X 2, and the formula to be used for calculating the electromotive force of a cell will be w-q^^» In applying this calculation to Daniell's and Grove's cells, we shall employ the following heats of combination, which are given on page 614 of Watts' "Dictionary of Chemistry," vol. vii., and are based on Julius Thomson's observations : — Zn, 0, S03, Aq., 108,462 Cu, O, S03, Aq., 54,225 N202, 03, Aq., 72,940 N202, 0, Aq., 36,340 In Daniell's cell, zinc is dissolved and copper is set free, we have, accordingly, H = 108,462 - 54,225 = 54,237. In Grove's cell, zinc is dissolved and nitric acid is 182 UNITS AND PHYSICAL CONSTANTS. [chap. changed into nitrous acid. The thermal value of this latter change can be computed from the third and fourth data in the above list, as follows : — 72,940 is the thermal value of the action in which, by the oxidation of one equivalent of N^O^ and combination with water, two equivalents of NHO^ (nitric acid) are produced. 36,340 is the thermal value of the action in which, by the oxidation of one equivalent of N^O^ and combination with water, two equivalents of NHO^ (nitrous acid) are produced. The difference 36,600 is accordingly the ther- mal value of the conversion of two equivalents of nitrous into nitric acid, and 18,300 is the value for the conversion of one equivalent In the present case the reverse changes take place. We have, therefore, H = 108,462 - 18,300 = 90,162. JH 19268 M82 X 108 for Danieirs cell. 1-965x108 „ Grove's „ These are greater by from 2 to 8 per cent, than the direct determinations given in § 205. 214. Examples in Electricity. 1. Two conducting spheres, each of 1 centim. radius, are placed at a distance of r centims. from centre to centre, r being a large number; and each of them is charged with an electrostatic unit of positive electricity. With what force will they repel each other ? Since r is large, the charge may be assumed to be uni- formly distributed over their surfaces, and the force will be the same as if the charge of each were collected at its centre. The force will therefore be -, of a dyne. Taking J as 4*2 x lO'^, the value of ^^^^-tto ^^^ ^® XI.I ELECTRICITY. 183 2. Two conducting spheres, eacli of 1 centim. radius, placed as in the preceding question, are connected one with each pole of a DanielFs battery (the middle of the battery being to earth) by means of two very fine wires whose capacity may be neglected, so that the capacity of each sphere when thus connected is sensibly equal to unity. Of how many cells must the battery consist that the spheres may attract each other with a force of - of a dyne, r being the distance between their centres in cen- tims. ? One sphere must be charged to potential 1 and the other to potential - 1. The number of cells required is ? = 535. •00374 3. How many DanielPs cells would be required to pro- duce a spark between two parallel conducting surfaces at a distance of '019 of a centim., and how many at a distance of -0086 of a centim. ? (See §§ 178, 184.) . 4-26 noQ 2-30 ^,^ •00374 ' -00374 4. Compare the capacity denoted by 1 farad with the capacity of the earth. The capacity of the earth in static measure is equal to its radius, namely 6*37 x 10^. Dividing by v'^ to reduce to magnetic measure, we have '71 x 10"^^, which is 1 farad multiplied by '71 x 10"', or is '00071 of a farad. A farad is therefore 1400 times the capacity of the earth. 5. Calculate the resistance of a cell consisting of a plate of zinc, A square centim s. in area, and a plate of copper of the same dimensions, separated by an acid 184 UNITS AND PHYSICAL CONSTANTS. [chap. solution of specific resistance 10^, the distance between the plates being 1 centim. Ana, — ) or — of an ohm. A A 6. Find the heat developed in 10 minutes by the passage of a current from 10 Daniell's cells in series through a wire of resistance 10^^ (that is, 10 ohms), assuming the electromotive force of each cell to be I'l X 10®, and the resistance of each cell to be 10^. Here we have Total electromotive force = 1 '1 x P. Resistance in battery = 10^^. Resistance in wire = 10^^. M xlOQ 2 X lO^o Current = l'^ "! LT = -55 x lO'i = -055. Heat developed in ) ^ (-0552) x IQio ^ 7.2004 wire per second ) 4*2 x 10"^ ~ " * Hence the heat developed in 10 minutes is 43214 gramme-degrees. 7. Find the electromotive force between the wheels on opposite sides of a railway carriage travelling at the rate of 30 miles an hour on a line of the ordinary gauge [4 feet 8J inches] due to cutting the lines of force of terrestrial magnetism, the vertical intensity being '438. The electromotive force will be the product of the velocity of travelling, the distance between the rails, and the vertical intensity, that is, (44-7 X 30) (2-54 x 56-5) (-438) = 84,300 electromagnetic units. This is about of a volt 1 Ji\j\j XI.] ELECTRICITY. 185 8. Find the electromotive force at the instant of passing tlie magnetic meridian, in a circular coil consisting of 300 turns of wire, revolving at the rate of 10 revolutions per second about a vertical diameter ; the diameter of the coil being 30 centims., and the horizontal intensity of terrestrial magnetism being '1794, no other magnetic influence being supposed present. Self-induction can be left out of account, because the current is a maximum. The numerical value of the lines of force which go through the coil when inclined at an angle 6 to the meridian, is the horizontal intensity multiplied by the area of the coil and by sin 6 ; say nH-ira^ sin 6, where H = *1794, a =15, and 7i = 300. The electromotive force at any instant is the rate at which this quantity increases or diminishes ; that is, TiKira^ cos ^ . w, if w denote the angular velocity. At the instant of passing the meridian cos ^ is 1, and the electromotive force is nUTra^d), With 10 revolutions per second the value of cd is 27r X 10. Hence the electromotive force is •1794 X (3142)2 X 225 x 20 x 300 = 2-39 x 10^. This is about t^ of a volt. 42 190. To investigate the magnitudes of units of length, mass, and time which will fulfil the three following conditions : — 1. The acceleration due to the attraction of unit mass at unit distance shall be unity. 2. The electrostatic units shall be equal to the electro- magnetic units. 186 UNITS AND PHYSICAL CONSTANTS. [chap. 3. The density of water at 4" C. shall be unity. Let the 3 units required be equal respectively to L centims., M gi*ammes, and T seconds. We have in C.G.S. measure, for the acceleration due to attraction (§ 72), acceleration = ,— r^,, where C = 6*48 x 10"^ : (distance)^ and in the new system we are to have mass acceleration = (distance)^* Hence, by division, acceleration in C.G.S. units acceleration in new units _p mass in C.G.S. units (distance in new units)^ mass in new units (distance in C.G.S. units)^ ' , , . . L ^M that is, _ = (J . vxA^v x«, 1.2 This equation expresses the first of the three conditions. The equation =^=v expresses the second, v denoting 3 X 1010. The equation M = L^ expresses the third. Substituting L^ for M in the first equation, we find T= ^/^. Hence, from the second equation, and from the third, H'4)' XI.] ELECTRICITY. 187 Introducing the actual values of C and v, we have approximately T = 3928, L= M78 x IQi*, M = 1-63 x 10*2 ; that is to say, The new unit of time will be about P 5 J™ ; The new unit of length will be about 118 thousand earth quadrants ; The new unit of mass will be about 2*66 x 10^* times the earth's mass. Electrodynamics, 191. Ampere's formula for the repulsion between two elements of currents, when expressed in electromagnetic units, is cc' ds . ds' r 2 (2 sin a sin a cos - cos <x cos a ), where c, c' denote the strengths of the two currents ; dsy ds' the lengths of the two elements ; a, a' the angles which the elements make with the line joining them ; r the length of this joining line ; the angle between the plane of r, ds^ and the plane of r, ds'. For two parallel currents, one of which is of infinite length, and the other of length Z, the formula gives by integration an attraction or repulsion, 21 , where D denotes the perpendicular distance between the currents. 1 88 UNITS AND PHYSICAL CONSTANTS, [chap. xi. Eocample, Find the attraction between two parallel wires a metre long and a centim. apart when a current of - - is passing through each. Here the attraction will be sensibly the same as if one of the wires were indefinitely increased in length, and will be 200/ !00/iy_2. that is, each wire will be attracted or repelled with a force of 2 dynes, according as the directions of the currents are the same or opposite. 189 OMISSION (to be added to § 63, p. 61). According to experiments by Quincke (Berlin Transac- tions, April 5, 1885) the following are the compressions due to the pressure of one atmosphere. They are ex- pressed in millionths of the original volume : — Compression in miilionths. ^ '^ , at 0° C. at <• C. t. Glycerine, 25*24 25*10 19*00 Rape oil (rttbol) 48*02 58*18 17*80 AhnondoU 48*21 56*30 19*68 OUveoil, 48*59 61*74 18*3 Water, 50*30 45*63 22*93 Bisulphide of carbon, 53 *92 63 *78 17 *00 Oil of turpentine, 58*17 77*93 18*56 Benzol from benzoic acitl, — 66*10 16*78 Benzol, — 62*84 16*08 Petroleum, 64*99 74*50 19*23 Alcohol, 82*82 95*95 17*51 Ether, 115*57 147*72 21*36 CORRECTION (p. 84). Benoit's results on refraction of air will not appear in vol. v., but in a later volume. 190 SUGGESTION FOR WRITING DECIMAL MULTIPLES AND SUBMULTIPLES. Professor Newcomb has suggested, as a possible improve- ment in future editions of this work, the employment of powers of 1000 instead of powers of 10 as factors (a plan which corresponds with the usual division of digits into ))eriods of 3 each)^ and the employment of the letter m in this connection to denote 1000. Thus, instead of 1*226 x 10^, we should write 122*6 m. 1-006x107, „ 10 06 m2. •000 000 9, „ -9 7n-\ The plan appears to possess some advantages ; and if the symbol m for 1000 is not sufficiently self-explanatory, we might write 122*6 x lO^, 1006 x 10«, -9 x 10"^ We place the suggestion on record here that it may not be overlooked. 191 APPENDIX. ^rst Report oftlte Committee for the Selection aiid Nomenclature of DynamicaX and ElectHcaZ UnitSy the Committee consisting of Sm W. Thomson, F.R.S., Professor G. C. Foster, F.R.S., Professor J. C. Maxwell, F.R.S., Mr. G. J. Stoney, F.R. S. ,* Professor Fleeming «Tenkin, F.R. S. , Dr. Siemens, F.R.S., Mr. F. J. Bramwell, F.R.S., and Professor Everett (Reporter). We consider that the most urgent portion of the task intrusted to us is that which concerns the selection and nomenclature of units of force and energy ; and under this head we are prepared to offer a definite recommendation. A more extensive and difficult part of our duty is the selection and nomenclature of electrical and magnetic units. Under this head we are prepared with a definite recommendation as regards iselection, but with only an interim recommendation as regards nomenclature. Up to the present time it has been necessary for every person who wishes to specify a magnitude in what is called ** absolute " measure, to mention the three fundamental units of mass, length, and time which he has chosen as the basis of his system. This necessity will be obviated if one definite selection of three funda- mental units be made once for all, and accepted by the general -consent of scientific men. We are strongly of opinion that such a selection ought at once to be made, and to be so made that there will be no subsequent necessity for amending it. We think that, in the selection of each kind of derived unit, all arbitrary multiplications and divisions by powers of ten, or other factors, must be rigorously avoided, and the whole system of * Mr. Stoney objected to the selection of the centimetre as the unit of length. 192 APPENDIX. fundamental units of force, work, electrostatic, and electromag- netic elements must be fixed at one common level — that level, namely, which is determined by direct derivation from the three fundamental units once for all selected. The carrying out of this resolution involves the adoption of some units which are excessively large or excessively small in comparison with the magnitudes which occur in practice ; but a remedy for this inconvenience is provided by a method of denoting decimal multiples and sub-multiples, which has already been extensively adopted, and which we desire to recommend for. general use. On the initial question of the particular units of mass, length, and time to be recommended as the basis of the whole system, a protracted discussion has been carried on, the principal point discussed being the claims of the gramme, the metre^ and the second, as against the gramme, the centimetre^ and the second, — the former combination having an advantage as regards the simplicity of the name mttrey while the latter combination has the advantage of making the unit of mass practically identical with the mass of unit- volume of water — in other words, of making the value of the density of water practically equal to unity. We are now all but unanimous in regarding this latter element of simplicity as the more important of the two ; and in support of this view we desire to quote the authority of Sir W. Thomson, who has for a long time insisted very strongly upon the necessity of employing units which conform to this condition. We accordingly recommend the general adoption of the Centi- metre, the Gramme, and the Second as the three fundamental units ; and until such time as special names shall be appropriated to the units of electrical and magnetic magnitude hence derived, we recommend that they be distinguished from ** absolute" units otherwise derived, by the letters "C.G.S." prefixed, these being the initial letters of the names of the three fundamental units. Special names, if short and suitable, would, in the opinion of a majority of us, be better than the provisional designations ** C.G.S. unit of . . . ." Several lists of names have already been suggested ; and attentive consideration will be given to any further APPENDIX. 193 suggestions which we may receive from persons interested in electrical nomenclature. The "ohm," as represented by the original standard coil, is approximately 10» C.G.S. units of resistance; the "volt" is approximately lO^ C.G.S. units of electromotive force ; and the " farad " is approximately -^ of the C.G.S. unit of capacity. For the expression of high decimal multiples and sub-multiples, we recommend the system introduced by Mr. Stoney, a system which has already been extensively employed for electrical pur- poses. It consists in denoting the exponent of the power of 10, which serves as multiplier, by an appended cardinal num- ber, if the exponent be positive, and by a prefixed ordinal number if the exponent be negative. Thus 10^ grammes constitute a gramme-nine; — ^ of a gramme 10^ constitutes a ninth-gramme; the approximate length of a quadrant of one of the earth's meridians is a metre-seven, or a centimetre- nine. For multiplication or division by a million, the prefixes mega* and micro may conveniently be employed, according to the present custom of electricians. Thus the megohm is a million ohms, and the microfarad is the millionth part of a farad. The prefix mega is equivalent to the affix six. The prefix micro is equivalent to the prefix sixth. The prefixes ktlo, hecto, deca, deci, centi, milli can also be em- ployed in their usual senses before all new names of units. As regards the name to be given to the C.G.S. unit of force, we recommend that it be a derivative of the Greek diJva/us. The form dyrumny appears to be the most satisfactory to etymologists. Dynam is equally intelligible, but awkward in sound to English ears. The shorter form, dyne, though not fashioned according to strict rules of etymology, will probably be generally preferred in this country. Bearing in mind that it is desirable to construct a system with a view to its becoming international, we think that * Before a vowel, either tmq or megcUt as euphony may suggest, may be employed instead of mega. N 194 APPENDIX. the termmation of the word should for the present be left an open question. But we would earnestly request that, whichever form of the word be employed, its meaning be strictly limited to the unit of force of the C.G.S. system — that is to say, the force which, acting upon a gramme of matter for a second, generates a velocity of a centimetre per second. The C.G.S. unit of work is the work done by this force working through a centimetre; and we propose to denote it by some deriva- tive of the Greek fpyov. The forms ergon, ergal, and erg have been suggested; but the second of these has been used in a different sense by Clausius. In this case also we propose, for the present, to leave the termination unsettled; and we request that the word ergon, or erg, be strictly limited to the C.G.S. unit of work, or what is, for purposes of measurement, equivalent to this, the C.G.S. unit of energy, energy being measured by the amount of work which it represents. The C.G.S. unit of power is the power of doing work at the rate of one erg per second ; and the power of an engine, under given conditions of working, can be specified in ergs per second. For rough comparison with the vulgar (and variable) units based on terrestrial gravitation, the following statement will be useful : — The weight of a gramme, at any part of the earth's surface, is about 980 dynes, or rather less than a kilodyne. The weight of a kilogramme is rather less than a megadyne, being about 980,000 dynes. Conversely, the dyne is about 1 *02 times the weight of a milli- gramme at any part of the earth's surface ; and the megadyne is about 1 *02 times the weight of a kilogramme. The kilogrammetre is rather less than the ergon-eight, being about 98 million ergs. The gramme-centimetre is rather less than the kilerg, being about 980 ergs. For exact comparison, the value of g (the acceleration of a body falling in vacuo) at the station considered must of course be known. In the above comparison it is taken as 980 C.G.S. units of acceleration. APPENDIX. 195 One horse-potoer is about three quarters of an erg-ten per second. More nearly, it is 7 '46 erg-nines per second, and one force-de-cheval is 7 '36 erg-nines per second. The mechanical equivalent of one gramme-degree (Centigrade) of heat is 41*6 megalergs, or 41,600,000 ergs. Second Report oftlie Committee for the Selection and Nomeiidature of Dynamical and Electrical Units, the Committee consisting of Professor Sir W. Thomson, F.R.S., Professor G. C. Foster, F.R.S., Professor J. Clerk Maxwell, F.R.S., G. J. Stone Y, F.R.S., Professor Fleeming Jenkin, F.R.S., Dr. C. W. Siemens, F.R.S., F. J. Bramwell, F.R.S., Professor W. G. Adams, F.R.S., Professor Balfour Stewart, F.R.S., and Professor Everett (Secretary). The Committee on the Nomenclature of Dynamical and Electrical Units have circulated numerous copies of their last year's Report among scientific men both at home and abroad. They believe, however, that, in order to render their recom- mendations fully available for science teaching and scientific work, a full and popular exposition of the whole subject of physical units is necessary, together with a collection of examples (tabular and otherwise) illustrating the application of systematic units to a variety of physical measurements. Students usually find peculiar difficulty in questions relating to units; and even the experienced scientific calculator is glad to have before him con- crete examples with which to compare his own results, as a security against misapprehension or mistake. Some members of the Committee have been preparing a small volume of illustrations of the C.G.S. system [Centimetre-Gramme- Second system] intended to meet this want. [The first edition of the present work is the volume of illustra- tions here referred to.] 196 INDEX. The numbers refer to the pages. Acceleration, 25. Acoustics, 70-74. Adiabatic compressioD, 125. Air, collected aata for, 129. , density of, 43. , expansion of, 99. , specific heat of, 94, 123. , tnermal conductivity of, 108. Ampere as unit, 151-153. Ampere's formula, 187. Aqueous vapour, pressure of, 100-102. , density of, 102. Astronomy, 65-69. Atmosphere, standard, 42, 43. , its density upwards, 47. Atomic weights, 180. Attraction, constant of, 67. at a point, 17. Angle, 16. , solid, 17. Barometer, correction for capil- larity, 51. Barometric measurements of heights, 47. pressure, 42. Batteries, 166-168, 172, 181. Boiling points, 98. of water, 100-102. Boyle's law, departures from, 99. Bullet, melted Dy impact, 31. Candle, standard, 86. Capacity, electrical, 141-143. , specific inductive, 147-150. , thermal, 87-95. Capillarity, 49-51. Carcel, 86. Cells, 166-168, 172, 181. Centimetre, reason for selecting, 23, 192. Centre of attraction, strength of, 17. Centrifugal force, 32. at equator, 34. C.G.S. system, 23, 192. Change of volume in evapora- tion, 97. in melting, 96, 97. Change-ratio, 9. Chemical action, heat of, 122. equivalents, 180. Clark's standard cell, 168. Cobalt, magnetization of, 136. Coil, revolving, 185. Combination, heat of, 122, 181. Combustion, heat of, 122. Common scale needed, 22. Comparison of standards (IlVench and English), 1, 2. INDEX. 197 Compressibility of liquids, 60, 61, 189. of solids, 61-63. Compression, adiabatic, 125. Conductivity (thermal) defined, 103. , thermometric and calori- metric, 105. of air, 108. of liquids, 116, 117. of various solids, 109-116. Congress of electricians, 153. Contact electricity, 168-172. Cooling, 117-120. Current, beat generated by, 143, 166. , unit of, 141, 142, 151, 153. Curvature, dimensions of, 17, 18. DanieU's cell, 166, 167, 173, 181. Day, sidereal, 66. Decimal multiples, 24, 190, 193. Declination, magnetic, at Green- wich, 138. Densities, table of, 40. of gases, 44. of water, 38-39. Density as a fundamental unit, 146. Derived units, 5, 6. Dew-point from wet and dry bulb, 102. Diamagnetic substances, 133. Diamond, specific heat of, 90. Diffusion, coefficient of, 105-108. Diffusivity (thermal), 105. Dimensionid equations, 9, 34-37. Dimensions, 7-9, 34-37. Dip at Greenwich, 138. Dispersive powers of gases, 83- 85. of solids and liquids, 77-82. Diversity of scales, 22. "Division," extended sense of, 10. Doable refraction, 81. Dynamics, 15-17. Dyne, 27, 193. Earth as a magnet, 136. , size, figure, and mass of, 65. Elasticity, 52-64. , effected by heat of com- pression, 127. Electric units, tables of their dimensions, 143, 146. Electricity, 140-148. Electrochemical equivalents, 180. Electrodynamics, 187. Electromagnetic units, 142. Electromotive force, 166-172, 180182. Electrostatic units, 140. Emission of heat, 117-120. Energy, 29. , dimensions of, 16. Equations, dimensional, 9, 34-37. , physical, 12. Equivalent, mechanical, of heat, 120. Equivalents, electrochemical, 180. Erg, 29, 194. Evaporation, change of volume in, 97. Examples in electricity, 1 82-1 86. in theory of units, 12-15, 34-37. in magnetism, 134-137. Expansion of gases, 99. of mercury, 128, 129. of various substances, 128, 129. Extended sense of ''multiplica- tion " and "division, 10. Farad, 151-153. compared with earth, 183. Field, intensit^r of, 131. Films, tension in, 49, 50. , thickness of, 50, 51. 198 INDEX. Foot-pound and foot-poundal, 30. Force, 27. , dimensions of, 15. at a point, 17. , various units of, 4. Freezing-point, change with pressure, 124. Frequencies of luminous vibra- tions, 77. Fundamental units, 6. , choice of, 19. reduced to two, 68. Gases, densities of, 44. , expansion of, 99. , indices of refraction of, 82. , inductive capacities of, 151. , two specific heats of, 123. Gauss's expression for magnetic potential, 138. pound-magnet, 134. units of intensity, 137. Geometrical quantities, dimen- sions of, 15-18. Gottingen, total intensity at, 137. Gramme-degree (unit of heat), 88. Gravitation in astronomy, 67. Gravitation measure of force and work, 28, 30. Gravity, terrestrial, 25-27. Greenwich, magnetic elements at, 138. Grove's ceU, 167, 172, 181. Heat, 87-130. generated by current, 143, 166. , mechanical equivalent of, 120. * of combination, 122, 181. of compression, 125. ., unit of, 87, 88. — , various units of, 3. Height, measured by barometer, 47. Homogeneous atmosphere,45-47. Horse-power, 30. Hydrostatics, 38-51. Hypsometric table of boiling points, 100. Ice, specific gravity of, 96, 125. , specific heat of, 92. , electrical resistance of, 163. Indices of refraction, 77-85. related to induc- tive capacities, 147, 148. Inductive capacity, 147-150. Induction, magnetic, coefficient of, 133. Insulators, resistance of, 164, 165. Interdiffusion, 106-108. Joule's equivalent, 120. Kilogramme and pound, 2. Kinetic energy, 29. KupfFer's determination of den- sity of water, 38. Large numbers, mode of expres- sing, 24, 190, 193. Latent heats, 95-98. Latimer Clark's cell, 168. Light, 75-86. , velocity of, 75, 76. , wave-lengths of, 76. Magnetic elements at Green- wich, 138. susceptibility, 133. units, 131, 132. Magnetism, 131-139. , terrestrial, 136-138. Magneto-optic rotation, 139. Magnetization, intensity of, 132, 133. Mass, standards of, 20. INDEX. 199 Mechanical equivalent of heat, 120. quantities, dimensions of, 15. units, 27. Mega, as prefix, 42, 193. Melting points, 95-97. Metre and yard, 1. Micro as prefix, 193. Microfarad, 151. Moment of couple, 16. of inertia, 16. of magnet, 132. of momentum, 16. Momentum, 15. Moon, 66. * 'Multiplication, "extended sense of, 10. Neutral point (thermoelectric), 175. Newcomb on decimal multiples, 190. Nickel, magnetization of, 136. Numerical value, 5. Ohm as unit, 151-154. earth quadrant per second, 155. -, ** legal," 153. Optics, 75-86. Paramagnetic substances, 133. Pendulum, seconds', 25, 26. ** Per," meaning of, 10. Physical deductions from di- mensions, 34-37. Platinoid, 160. Platinum, specific heat of, 90. Poisson's ratio, 62. Potential, electric, 140. , magnetic, 131. Poundal, 28. Powers of ten as factors, 24, 190, 193. Pressure, dimensions of, 17. of liquid columns, 42. Pressure, various units of, 3, 4. Pressures of vapours, 101. Pressure-height, 46. Quantity of electricity, 140, 142. Radian, 16. Radiation, 117-120. Ratios of two sets of electric units, 143. Refraction, indices of, 77-85. Reports of Units Committee, 191-195. Resilience, 54. affected by heat of com- pression, 126. Resistance, electrical, 158-166. of a cell, 161, 183. of wires, 165, 166. Rigidity, simple, 55. Rotating coil, 185. Saturation, magnetic, 133-136. Shear, 55-58. Shearing stress, 58-60. Siemens^ unit, 152-154. Soap films, 50. Sound, faintest, 74. , velocitv of, 70-73. Spark, length of, 155-157. Specific gravities, 40. , heat, 88-95. , two, of gases, 123. , inductive capacity, 147- 150. Spring balance, 31. Standards, French and English, 1,2. of length, 21. of mass, 20. of time, 21. Steam, pressure and density of, 100-102. , total and latent heat of, 98. Stoney's nomenclature for multiples, 193. 200 INDEX. Strain, 52, 53, 55-58. , dimensions of, 53. Stress, 52-54, 58-60. , dimensions of, 54. Strings, musical, 73. Sun's distance and parallax, 66. Supplemental section on dimen- sions, 34-37. Surface-conduction, 1 17-120. Surface-tension, 49, 50. Telegraphic cables, resistance of, 160. Tenacities, table of, 64. Tensions of liquid surfaces, 49, 50. Thermodynamics, 120-128. Thermoelectricity, 172-178. Time, standard of, 21. Tortuosity, 17. Two fundamental units suffici- ent, 68. Unit, 5. Units, derived, 5, 6. , dimensions of, 7-9. , special problems on, 69, 185. Vapours, pressure of, 101. Velocity, 6, 9. of Ught, 75, 76. of sound, 70-73. , various units of, 2. Vibrations per second of light, 77. Volt, 151-153. Volume, by weighing in water, 40. of a gramme of gas, 44. , unit of, 5. Volume resilience, 55, 60-63, 189. Water, compressibility of, 60, 61, 189. , densitjr of, 38, 39. , expansion of, 39. , specific heat of, 87, 88. , weighing in, 40. ii Watt (rate of working), 4, 30. Weight, force, and mass, 27, 28. , standards of, 20-2. Wires (Imperial gauge), 165,166. Work, 29, 30, 3, 4. , dimensions of, 16. done by current, 143. Working, rate of, 30, 3, 4. Year, sidereal and tropical, 66. Young's modulus, 55. PBINTED BT ROBERT MACLEHOSE, UNIVERSITY PRESS, QLASaOW. r A Catalogue OF WORKS ON Mathematics, Science, ANX> History and Geography. PUBLISHBD BY Macmillan & Co., Bedford Street, Strand, London. lo. 4. 6 CONTENTS. Arithmbtic . • • » . • ■ Algebra •••••• 4 Euclid, and Elbmkntary Gbombtry . 5 MbNSV RATION ..... . 6 Higher Mathematics . • . • . . 6 SCZENCB— Na'a'URAL Philosopht . • • • • >4 Astronomy ...•.., 19 C4iBMISTRY ...•.• 30 lilOLOGY tl Medicine ..••••, as Anthropology ....•, . i6 Physical Geography, and Geology . >6 Agricolturb ...... «7 •PouTicAL Economy . . . . a3 Mental and Moral Phiix>sophy . I . * HISTORY AND GEOGRAPHY * 29 MATHEMATICS. (r) Arithmetic, (2) Algebra, (3) Euclid and Elc mentary Geometry, (4) Mensuration, (5) Highe Mathematics. ARITHMETIC. Aldis. — THE GREAT GIANT ARITHMOS. A most Elementa Arithmetic for Childn n. By Mary Steadman Aldis. Wi Illustrations. Globe 8vo. 2s. 6c/. Brook-Smith (J.).— ARITHMETIC IN THEORY AN PRACTICE. By J. 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